id int64 | problem_number string | title string | statement string | background string | difficulty_level_id int64 | status string | proposed_by string | proposed_year int64 | category_id int64 | set_id int64 | view_count int64 | favorite_count int64 | created_at timestamp[s] | updated_at timestamp[s] | published bool | category dict | difficulty dict | set dict | duplicates list | status_review unknown | archive_reason string | duplicate_of list | old_problem_number string | notes string | tags list | literature_assessment string | literature_sources list | literature_checked_at timestamp[s] | solution_reference string | source_url string | source string | prize string | references list | clean_statement string | original_statement string | statement_status string | statement_verification string | research_classification string | research_summary string | research_difficulty_suggested string | historical_research_summary string | research_summary_reviewed_at timestamp[s] | source_citation string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
800,370 | AMR-125-0008 | Coxeter Groups and Unimodality — Conjecture 8 | For $\sigma \in S_n$ let
\[
L(\sigma) := | \{ (i,j) \in [n]^2 : i<j, \sigma(i)>\sigma(j), \; i \not \equiv j \pmod{2} \} |.
\]
The statistic $L$ is known as the *odd length* or *odd inversion number*
of $\sigma$. The statistic was introduced in [KloVol] in relation to formed spaces
and has been further studied in [BreC... | Source list: Brenti - Some Open Problems on Coxeter Groups and Unimodality (2024)
Source item: Conjecture 8
Source URL: https://arxiv.org/abs/2410.09897
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Brenti | 2,024 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,371 | AMR-125-0009 | Coxeter Groups and Unimodality — Conjecture 9 | So, knowledge of the $ \widetilde{R}$-polynomials is equivalent to knowledge of
the $R$-polynomials. Combinatorial interpretations of the polynomials
$\{ \widetilde{R}_{u,v}(t) \}_{u,v \in W}$ have been given in [DeoInv] and
[Dye93], see [BB, Thms 5.3.4 and 5.3.7].
It is easy to see that, if $u \leq v$, then $deg( \wid... | Source list: Brenti - Some Open Problems on Coxeter Groups and Unimodality (2024)
Source item: Conjecture 9
Source URL: https://arxiv.org/abs/2410.09897
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Brenti | 2,024 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,372 | AMR-125-0010 | Coxeter Groups and Unimodality — Problem 10 | The following problem was
raised in [BreChro, Prob. 7.1].
Does $\tau(G;x)$ have only real roots for all graphs $G$? | Source list: Brenti - Some Open Problems on Coxeter Groups and Unimodality (2024)
Source item: Problem 10
Source URL: https://arxiv.org/abs/2410.09897
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Brenti | 2,024 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,373 | AMR-125-0011 | Coxeter Groups and Unimodality — Problem 11 | Let $\sum_{i=0}^{d} h_i t^i \in \mathbb{N}[t]$ be a symmetric, monic polynomial with only real roots. Is there a simplicial convex polytope whose $h$-vector is $(h_0, \ldots , h_d)$? | Source list: Brenti - Some Open Problems on Coxeter Groups and Unimodality (2024)
Source item: Problem 11
Source URL: https://arxiv.org/abs/2410.09897
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Brenti | 2,024 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,374 | AMR-125-0012 | Coxeter Groups and Unimodality — Problem 12 | Let $\sum_{i=0}^{d} h_i t^i \in \mathbb{N}[t]$ be a polynomial with only real roots
such that $h_0=1$ and $h_0 < h_1 < \cdots < h_k$ for some $0 \leq k \leq d$. Is there a simplicial complex whose $f$-vector is $(h_0, h_1-h_0, \ldots , h_k-h_{k-1})$? | Source list: Brenti - Some Open Problems on Coxeter Groups and Unimodality (2024)
Source item: Problem 12
Source URL: https://arxiv.org/abs/2410.09897
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Brenti | 2,024 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,375 | AMR-125-0013 | Coxeter Groups and Unimodality — Conjecture 13 | Let $(W,S)$ be a Coxeter system. It is of interest, and difficult, to obtain
properties of the
rank generating function of Bruhat intervals $[u,v]$, $u,v \in W$ (see, e.g., [BjoEke]).
In this respect, I feel that the following holds.
Let $W$ be a Weyl group, and $u,v \in W$. Then $[u,v]$ is rank log-concave. | Source list: Brenti - Some Open Problems on Coxeter Groups and Unimodality (2024)
Source item: Conjecture 13
Source URL: https://arxiv.org/abs/2410.09897
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Brenti | 2,024 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,376 | AMR-125-0014 | Coxeter Groups and Unimodality — Conjecture 14 | Many matrices arising in combinatorics
are known to be TP (see, e.g., [BreMem], [BreCombTP], and
[BreTPComb]). It is therefore surprising that for a fundamental and
old combinatorial matrix such as the one consisting of the Eulerian numbers this property has not yet been settled.
For $n \in \mathbb{P}$ and $k \in \math... | Source list: Brenti - Some Open Problems on Coxeter Groups and Unimodality (2024)
Source item: Conjecture 14
Source URL: https://arxiv.org/abs/2410.09897
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Brenti | 2,024 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,377 | AMR-125-0015 | Coxeter Groups and Unimodality — Conjecture 15 | For all $i,j,r \in \mathbb{N}$
the determinant of the submatrix determined by the rows indexed by
$i,i+1, \ldots , i+r$ and columns indexed by $j,j+1, \ldots , j+r$
is a monotonically increasing function of $i \in \mathbb{N}$. | Source list: Brenti - Some Open Problems on Coxeter Groups and Unimodality (2024)
Source item: Conjecture 15
Source URL: https://arxiv.org/abs/2410.09897
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Brenti | 2,024 | 2 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,378 | AMR-126-0001 | Open Problems in Arithmetic Dynamics — Question 1 | Let $S$ be a set, and let $T\subseteq S$ be a subset. Classify those maps
$f:S\to S$ with the property that for every $\alpha\in S$, the set
\[
\bigl\{ n \ge 0 : f^n(\alpha) \in T \bigr\}
\]
has some nice structure, for example, it is the union of finitely many
one-sided arithmetic progressions. | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 1
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,379 | AMR-126-0002 | Open Problems in Arithmetic Dynamics — Conjecture 2 | In view of Northcott's theorem, it is natural to inquire as to the
possible size of the finite set $\operatorname{PrePer}\bigl(f,\mathbb{P}^N(K)\bigr)$, which
leads to the following fundamental conjecture.
Fix integers $N\ge1$, $d\ge2$, and $D\ge1$. There is a constant
$C(N,d,D)$ such that for all degree $d$ morphisms... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 2
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Dynamical Uniform Boundedness Conjecture:... | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,380 | AMR-126-0003 | Open Problems in Arithmetic Dynamics — Conjecture 3 | \textup{(Poonen, 1998, [poonen:uniformboundrefined])}
For all $c\in\mathbb{Q}$, the set of periodic points \text{$\operatorname{Per}(x^2+c,\mathbb{Q})$} contains
no points of period strictly greater than $3$. | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 3
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,381 | AMR-126-0004 | Open Problems in Arithmetic Dynamics — Conjecture 4 | [Manes, 2008, [MR2407816]]
A $\mathbb{Q}$-rational periodic point of a map of the form $bx+cx^{-1}$ with
$b,c\in\mathbb{Q}^*$ has period $1$, $2$, or $4$. Further,
\[
\sup_{b,c\in\mathbb{Q}^*} \#\operatorname{PrePer}\bigl(bx+cx^{-1},\mathbb{P}^1(\mathbb{Q})\bigr) = 12.
\] | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 4
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,382 | AMR-126-0005 | Open Problems in Arithmetic Dynamics — Conjecture 5 | [Benedetto, Chen, Hyde, Kovacheva, White, 2014, {[MR3277939]}; Blanc--Canci--Elkies, 2015, [MR3431627]]
Let $f\in\operatorname{End}_2^1(\mathbb{Q})$ be a rational map of degree $2$ defined over $\mathbb{Q}$.
\begin{parts}
(a)
Let $P\in\mathbb{P}^1(\mathbb{Q})$ be periodic for $f$. Then $P$ has period at most $7$.
(b... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 5
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,383 | AMR-126-0006 | Open Problems in Arithmetic Dynamics — Conjecture 6 | [Poonen, 2012, {[arxiv1206.7104, Question 3.2.]}]
Let $K$ be a number field, let $X$ and $S$ be varieties defined
over $K$, let $\pi:X\to S$ be a morphism, and let $f:X\to X$ be an
$S$-morphism, which means that $\pi\circ f=\pi$. For each point $s\in
S({\bar K})$, the map $f$ restricted to the fiber $X_s:=\pi^{-1}(s)$... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 6
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,384 | AMR-126-0007 | Open Problems in Arithmetic Dynamics — Question 7 | \begin{parts}
(a)
For which choices of fields $K$, maps $f$, and basepoints $P$ is the
Odoni index $\iota_K(f,P)$ equal to $1$, i.e., when is the arboreal
representation $\rho^{\mathrm{arb}}_{f,P}$ surjective?
(b)
For which choices of fields $K$, maps $f$, and basepoints $P$ is the
Odoni index $\iota_K(f,P)$ finite, ... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 7
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,385 | AMR-126-0008 | Open Problems in Arithmetic Dynamics — Conjecture 8 (Odoni, 1985, [MR805714]) | Let $K$ be a Hilbertian field, (A field $K$ is
*Hilbertian* if it is not possible to cover $\mathbb{P}^1(K)$ by a
union of finitely manys sets of the form $F\bigl(C(K)\bigr)$,
where $C/K$ is a smooth curve and $F:C\to\mathbb{P}^1$ is a finite map of
degree at least $2$. For example, all number fields are Hilbertian,
bu... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 8
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Odoni, 1985, [MR805714] | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,386 | AMR-126-0009 | Open Problems in Arithmetic Dynamics — Conjecture 9 | Let $K$ be a field of characteristic $0$,
and let $f:\mathbb{P}^N\to\mathbb{P}^N$ be a morphism of degree $d\ge2$ defined
over $K$. Then for all sufficiently large $n$ we have (The
function $\mu$ is the M\"obius function, which we recall is defined
as follows: If $n$ is a product of $k$ distinct primes,
then $\mu(n)=(-... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 9
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,387 | AMR-126-0010 | Open Problems in Arithmetic Dynamics — Question 10 | Let $K$ be a field of characteristic $0$, let $f:\mathbb{P}^N\to\mathbb{P}^N$ be a
morphism of degree $d\ge2$ defined over $K$, and for $n\ge1$, let
$\nu=\nu(n,f)=\#\operatorname{Per}_n^*(f)$. Find general necessary and
sufficient conditions on $f$ for the following statements to
hold\textup:
\begin{parts}
(a)
The ind... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 10
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,388 | AMR-126-0011 | Open Problems in Arithmetic Dynamics — Conjecture 11 | [Dynamical Mordell--Lang Conjecture:
Bell, 2006, [MR2225492];
Denis, 1994, [MR1259107];
Ghioca--Tucker, 2009, [arxiv0805.1560]]
Let $X/\mathbb{C}$ be a quasi-projective variety, let $f:X\to X$ be a morphism,
let $P\in X$ be a point, and let $Y\subseteq X$ be a subvariety of $X$.
Then
\[
\{ n\ge 0 : f^n(P)\in Y \}
\]
i... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 11
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,389 | AMR-126-0012 | Open Problems in Arithmetic Dynamics — Conjecture 12 | [Amerik--Bogomolov--Rovinsky, 2011, [MR2862064];
Medvedev--Scanlon, 2009, [arxiv0901.2352]]
Let $X$ be a quasi-projective variety defined over an algebraically
closed field $K$ of characteristic $0$. Let $f:X\to X$ be an
endomorphism defined over $K$, and assume that $f$ is not a fibered
map.
\begin{parts}
(a) \mathr... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 12
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,390 | AMR-126-0013 | Open Problems in Arithmetic Dynamics — Conjecture 13 | [Ghioca--Tucker--Zhang, 2011, {[MR2854724, Conjecture 2.4]}]
Let $X/\mathbb{C}$ be a projective variety, let $f: X \to X$ be a polarizable
endomorphism, and let $Y$ be a
subvariety of $X$ which has no component contained entirely within the singular
locus of $X$. Then the following are equivalent\textup:
\begin{parts}... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 13
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,391 | AMR-126-0014 | Open Problems in Arithmetic Dynamics — Question 14 | Let $G_{d,m,n}(c)\in\mathbb{Z}[c]$ be the Gleason polynomial of $x^d+c$.
\begin{parts}
(a)
Is it true that $G_{2,m,n}$ is irreducible in $\mathbb{Q}[c]$ for all $m\ge0$
and $n\ge1$?
(b)
For which $(d,m,n)$ is $G_{d,m,n}$ irreducible in $\mathbb{Q}[c]$?
(c)
More generally, what is the complete factorization of $G_{d,... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 14
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,392 | AMR-126-0015 | Open Problems in Arithmetic Dynamics — Question 15 | We collect some of the natural questions that one might ask
about the moduli spaces $\mathcal{M}_d^N$ and about automorphism groups.
\begin{parts}
(a)
For $N\ge2$, are all \textup(some, none\textup) of the moduli
spaces $\mathcal{M}_d^N$ rational varieties?
(b)
For each $(N,d)$, describe the possible groups $\operato... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 15
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,393 | AMR-126-0016 | Open Problems in Arithmetic Dynamics — Question 16 | The fact that the bound for $\operatorname{FOM/FOD}(1,d)$ does not depend on $d$ suggests
the following question.
For a given $N\ge1$, is
\[
\sup_{d\ge2} \operatorname{FOM/FOD}(N,d) < \infty?
\]
If so, what is its value as a function of $N$? | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 16
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,394 | AMR-126-0017 | Open Problems in Arithmetic Dynamics — Conjecture 17 (Dynamical Andr{\'e}--Oort Conjecture) | \textup{(Baker--DeMarco, ${}\approx2011$, [arxiv1211.0255,MR2884382])}
Let $X\subseteq\mathcal{M}_d^{\mathrm{crit}}$ be an algebraic subvariety such that
the PCF maps in $X$ are Zariski dense in $X$. Then $X$ is cut out by
``*critical orbit relations*.'' | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 17
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Dynamical Andr{\'e}--Oort Conjecture | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,395 | AMR-126-0018 | Open Problems in Arithmetic Dynamics — Conjecture 18 (Dynamical Andr{\'e}--Oort Conjecture for 1-Dimensional Families) | \textup{(Baker--DeMarco, 2013, [arxiv1211.0255])}
Let $T$ be a non-singular algebraic curve, and let
\[
T \longrightarrow\mathcal{M}_d^{\mathrm{crit}},\quad
t \longmapsto (f_t,P_{1,t},\ldots,P_{2d-2,t})
\]
be a family of rational maps with marked critical points. Then the
following are equivalent\textup:
\begin{parts}
... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 18
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Dynamical Andr{\'e}--Oort Conjecture for... | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,396 | AMR-126-0019 | Open Problems in Arithmetic Dynamics — Question 19 | Find a ``nice'' function $C(N,d,D,s)$ so that every degree $d$
morphism $f:\mathbb{P}^N\to\mathbb{P}^N$ defined over a field $K$ of degree at
most $D$ and having at most $s$ primes of bad reduction satisfies
\[
\#\operatorname{PrePer}\bigl(f,\mathbb{P}^N(K)\bigr) \le C(N,d,D,s).
\] | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 19
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,397 | AMR-126-0020 | Open Problems in Arithmetic Dynamics — Question 20 | What is the exact value of $\operatorname{ShafDim}_d^N$\mathrm{?} In particular, is it true that
\[
\operatorname{ShafDim}_d^1=d+1\quad\text{for all $d\ge3$\mathrm{?}}
\] | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 20
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,398 | AMR-126-0021 | Open Problems in Arithmetic Dynamics — Question 21 | Give criteria, in terms of an integer $d\ge2$ and the geometry of a
portrait ${\mathcal{P}}$, that distinguish between the following two
situations\textup:
\begin{parts}
(1)
$\operatorname{\mathcal{G\!R}}_d^1[{\mathcal{P}}](K,S)/\operatorname{PGL}_2(R_S)$ is finite for all number fields $K$ and
all finite sets of prim... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 21
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,399 | AMR-126-0022 | Open Problems in Arithmetic Dynamics — Question 22 | Let $Y_0^{\mathrm{dyn}}(n)$ and $Y_1^{\mathrm{dyn}}(n)$ be the dynamical modular curves
classifying points, respectively cycles, of formal period $n$ attached
to the family of maps $x^2+c$, or $x^m+c$, or some other natural
$1$-parameter family of rational maps on $\mathbb{P}^1$ such as $bx+x^{-1}$.
\begin{parts}
(a)
... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 22
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,400 | AMR-126-0023 | Open Problems in Arithmetic Dynamics — Conjecture 23 (Orbit Density Conjecture for $\mathbb{A}^N$) | \textup{(Medvedev--Scanlon, 2009,
{[arxiv0901.2352, Conjecture 5.10]};
Amerik--Bogomolov--Rovinsky, 2011,
{[MR2862064, Conjecture 1.2]})}
Let $f:\mathbb{A}^N\to\mathbb{A}^N$ be a morphism defined over ${\bar{\mathbb{Q}}}$, i.e.,
$f=(f_1,\ldots,f_N)$, where the $f_i\in{\bar{\mathbb{Q}}}[x_1,\ldots,x_N]$ are
polynomials... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 23
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Orbit Density Conjecture for $\mathbb{A}... | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,401 | AMR-126-0024 | Open Problems in Arithmetic Dynamics — Conjecture 24 (Orbit Density Conjecture) | \textup{(Medvedev and Scanlon, 2009,
{[arxiv0901.2352, Conjecture 5.10]}; Amerik, Bogomolov and
Rovinsky, 2011, {[MR2862064, Conjecture 1.2]})}
Let ${\bar K}$ be an algebraically closed field of characteristic $0$,
let $X/K$ be a quasi-projective variety, and let $f:X\to X$ be a
dominant morphism. Assume that there do... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 24
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Orbit Density Conjecture | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,402 | AMR-126-0025 | Open Problems in Arithmetic Dynamics — Question 25 | Let $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map. Do
there exist a finite collection of constant coefficient linear
recursions
\[
L_n^{(1)},L_n^{(2)},\ldots,L_n^{(r)}
\]
and an index function $\iota:\mathbb{N}\to\{1,2,\ldots,r\}$ so that for all $n\ge1$ we
have
\[
\deg(f^n) = L_n^{(\iota_n)} ... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 25
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,403 | AMR-126-0026 | Open Problems in Arithmetic Dynamics — Conjecture 26 (Integrality Conjecture) | \textup{(Bellon--Viallet, [MR1704282], 1999)} Let
$f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map. Then the dynamical
degree $\delta_f$ of $f$ is an algebraic integer, i.e., $\delta_f$ is the root
of a monic polynomial having integer coefficients. | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 26
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Integrality Conjecture | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,404 | AMR-126-0027 | Open Problems in Arithmetic Dynamics — Question 27 | Let $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map. Is it
true that there exists an integer $\ell_f$ satisfying $0\le\ell_f\le
N$ such that (We recall that the notation $F(n)\asymp G(n)$
means that there are positive constants $c_1,c_2,c_3$, depending on $F$
and $G$, so that $c_1F(n) \le G(n) \l... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 27
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,405 | AMR-126-0028 | Open Problems in Arithmetic Dynamics — Question 28 | Let $X$ be a smooth projective variety of dimension $N$, and let
$f:X\dashrightarrow X$ be a dominant rational map. Is it true
that the higher order dynamical degrees $\delta_f^{(i)}$ are algebraic
integers for all $1\le i\le N$\mathrm{?} | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 28
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,406 | AMR-126-0029 | Open Problems in Arithmetic Dynamics — Question 29 | Let $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map defined
over ${\bar{\mathbb{Q}}}$, and let $P\in\mathbb{P}^N({\bar{\mathbb{Q}}})_f$ be a point whose
orbit $\mathcal{O}_f(P)$ is Zariski dense in $\mathbb{P}^N$. Is it true that
there exist integers $k_f$ and $\ell_f$ with $0\le\ell_f\le N$ such... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 29
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,407 | AMR-126-0030 | Open Problems in Arithmetic Dynamics — Question 30 | It is known that $\overline\mu_d(\mathbb{P}^N)\le d^{-(N-1)}$ for all $N\ge2$
and all $d\ge2$; see [arxiv0908.3835, Proposition 10]
and [MR2975153]. This leads to the obvious question.
What is the exact value of $\overline\mu_d(\mathbb{P}^N)$? | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 30
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,408 | AMR-126-0031 | Open Problems in Arithmetic Dynamics — Conjecture 31 (Dynamical Lang Height Conjecture) | Let $N\ge1$, $D\ge1$, and $d\ge2$. There are positive constants
$c_1(D,N,d)$ and $c_2(D,N,d)$ such that for all number fields $K/\mathbb{Q}$
with $[K:\mathbb{Q}]\le D$, all degree $d$ morphisms $f:\mathbb{P}^N\to\mathbb{P}^N$
defined over $K$, and all points $P\in\mathbb{P}^N(K)$ whose
orbit $\mathcal{O}_f(P)$ is Zaris... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 31
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Dynamical Lang Height Conjecture | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,409 | AMR-126-0032 | Open Problems in Arithmetic Dynamics — Question 32 | Given a family of maps $f:\mathbb{P}^N\to\mathbb{P}^N$ and a family of
points $P$, parametrized by $B$, describe the map
\[
B(K_v)\longrightarrow\mathbb{R}\cup\{\infty\},\quad
t \longmapsto \hat\lambda_{f_t,v}(P_t).
\]
In particular, to what extent is it a Weil function associated to some
divisor in $\operatorname{Div}... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 32
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,410 | AMR-126-0033 | Open Problems in Arithmetic Dynamics — Question 33 | Let $K(B)$ be the function field of an algebraic curve, let
$f:\mathbb{P}^N\to\mathbb{P}^N$ be a morphism of degree $d\ge2$ defined over the
field $K(B)$, and let $P\in\mathbb{P}^N\bigl(K(B)\bigr)$. Is the canonical
height ${\hat h}_f(P)$ always in $\mathbb{Q}$? If not, give necessary and
sufficient conditions for ${\h... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 33
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,411 | AMR-126-0034 | Open Problems in Arithmetic Dynamics — Conjecture 34 | Let $f\in\mathbb{C}_v(z)$ be a rational function of degree at least $2$, and
let ${\mathcal{J}}_f\subset\mathsf{P}_{\mathrm{Berk}}^1$ be the Julia set
of $f$. Then ${\mathcal{J}}_f\cap\mathbb{P}^1(\mathbb{C}_v)$ is equal to the closure
in $\mathbb{P}^1(\mathbb{C}_v)$ of the set of Type I repelling periodic points
of $f... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 34
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,412 | AMR-126-0035 | Open Problems in Arithmetic Dynamics — Question 35 | Let $f\in\mathbb{C}_v(z)$ with $\deg f = d\geq 2$. Can $f$ have infinitely many periodic cycles
of indifferent components that are not disks? If not, is there a bound in terms of $d$
for the number of such cycles? Do the answers depend on the field $\mathbb{C}_v$? | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 35
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,413 | AMR-126-0036 | Open Problems in Arithmetic Dynamics — Conjecture 36 | [[MR1781331]]
Let $K$ be a locally compact subfield of $\mathbb{C}_v$, and suppose that the
algebraic closure $\bar{K}$ of $K$ is dense in $\mathbb{C}_v$.
Let $f\in{K}(z)$ be a rational function of degree at least $2$.
Then $f$ has no wandering domains. | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 36
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,414 | AMR-126-0037 | Open Problems in Arithmetic Dynamics — Conjecture 37 | Let $K\subset\mathbb{C}_v$ be a discretely valued subfield whose algebraic
closure $\bar{K}$ is dense in $\mathbb{C}_v$. Let $p\geq 0$ be the residue
characteristic of $\mathbb{C}_v$, and let $f\in\mathbb{C}_v(z)$ be a rational
function of degree at least $2$. Suppose that there are no
points $\zeta\in\mathsf{P}_{\math... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 37
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,415 | AMR-126-0038 | Open Problems in Arithmetic Dynamics — Question 38 (Vague Motivating Question) | If we let $f$ range over $\operatorname{End}_{\mathsf{Set}}(S)$ or $\operatorname{Aut}_{\mathsf{Set}}(S)$ and let
$\#S\to\infty$, then there are known statistics for $\#\operatorname{Per}(f,S)$ and
for various natural quantities associated to the graph $\Gamma_f$ such
as its number of components, its largest component,... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 38
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Vague Motivating Question | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,416 | AMR-126-0039 | Open Problems in Arithmetic Dynamics — Conjecture 39 | \begin{parts}
(a)
Let $f:\mathbb{P}^N\to\mathbb{P}^N$ be a morphism of degree $d\ge2$ defined over
$\mathbb{Q}$, and let $P\in\mathbb{P}^N(\mathbb{Q})$ be a point whose $f$-orbit
$\mathcal{O}_f(P)$ is Zariski dense in $\mathbb{P}^N$. Then for every $\epsilon>0$, the
set of primes
\[
\bigl\{ p : \#\mathcal{O}_{\tilde f... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 39
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,417 | AMR-126-0040 | Open Problems in Arithmetic Dynamics — Question 40 | Let $f:\mathbb{P}^N\to\mathbb{P}^N$ be a morphism of degree $d\ge2$ defined over
$\mathbb{Q}$, and let $P\in\mathbb{P}^N(\mathbb{Q})$ be a point whose $f$-orbit
$\mathcal{O}_f(P)$ is Zariski dense in $\mathbb{P}^N$.
Under what conditions on $f$ is it true that for every $\epsilon>0$, the
set of primes
\[
\bigl\{ p : \#... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 40
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,418 | AMR-126-0041 | Open Problems in Arithmetic Dynamics — Question 41 | Fix an integer $N\ge1$ and a prime power $q$. Is it true that
\[
\frac{1}{\#\operatorname{End}_d^N(\mathbb{F}_q)} \sum_{f\in\operatorname{End}_d^N(\mathbb{F}_q)} \operatorname{Comp}(\Gamma_f) = \operatorname{Comp}_{\mathrm{avg}}\bigl(\#\mathbb{P}^N(\mathbb{F}_q)\bigr) + O(1)
\quad\text{as $d\to\infty$?}
\] | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 41
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,419 | AMR-126-0042 | Open Problems in Arithmetic Dynamics — Question 42 | Let $f:\mathbb{P}^N\to\mathbb{P}^N$ be a morphism of degree $d\ge2$ defined
over $\mathbb{Q}$. Is it true that for every $\epsilon>0$, the set of primes
\[
\left\{ p : \operatorname{Comp}(\Gamma_{\tilde f\bmod p}) \le \operatorname{Comp}_{\mathrm{avg}}(p^{(1-\epsilon)N})\right\}
\]
is a set of density $0$? Similarly wi... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 42
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,420 | AMR-126-0043 | Open Problems in Arithmetic Dynamics — Question 43 | Let $f(x)\in\mathbb{F}_q[x]$ be an \textup(irreducible\textup) polynomial,
let $g(x)\in\mathbb{F}_q[x]$, and for each $n\ge1$,
factor $g\bigl(f^n(x)\bigr)$ in $\mathbb{F}_q[x]$ as
\[
g\bigl(f^n(x)\bigr) = p_1(x)^{e_1} p_2(x)^{e_2} \cdots p_r(x)^{e_r}
\]
with $p_i(x)$ irreducible in $\mathbb{F}_q[x]$.
\begin{parts}
(a)... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 43
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,421 | AMR-126-0044 | Open Problems in Arithmetic Dynamics — Conjecture 44 | Let $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map defined
over $\mathbb{F}_q$.
\begin{parts}
(a)
$\mathbb{P}^N(\bar\mathbb{F}_q)_f$ is Zariski dense in $\mathbb{P}^N$.
(b)
$\displaystyle
\lim_{k\to\infty} \#\mathbb{P}^N(\mathbb{F}_{q^k})_f/\#\mathbb{P}^N(\mathbb{F}_{q^k}) = 1.
$
\end{parts} | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 44
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,422 | AMR-126-0045 | Open Problems in Arithmetic Dynamics — Conjecture 45 | \textup{(Jones--Levy, 2017, [MR3704363, Conjecture 1.2])} Let $K$
be a number field, let $f(x)\in K(x)$ be a rational function of degree
at least $2$, and let $\alpha\in K$ be a point that is not periodic
for $f$. Then $(f,\alpha)$ is eventually stable over $K$. | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 45
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,423 | AMR-126-0046 | Open Problems in Arithmetic Dynamics — Conjecture 46 | Let $K$ be a number field, let $f:\mathbb{P}^N\to\mathbb{P}^N$ be an endomorphism of
degree at least $2$ defined over $K$, and let $P\in\mathbb{P}^N(K)$ be a
point that is not periodic for $f$. Then $(f,P)$ is eventually stable
over $K$. | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 46
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,424 | AMR-126-0047 | Open Problems in Arithmetic Dynamics — Question 47 | Let $K$ be a number field, let $f:\mathbb{P}^N\to\mathbb{P}^N$ be an endomorphism of
degree at least $2$ defined over $K$, and let $X\subsetneq\mathbb{P}^N$ be a
subvariety that is defined over $K$ and irreducible over $K$.
\begin{parts}
(a)
If the iterated forward orbit $\bigcup_{n\ge1} f^n(X)$ of $X$ is
Zariski dens... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 47
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,425 | AMR-126-0048 | Open Problems in Arithmetic Dynamics — Question 48 | We recall that $\operatorname{End}_d^1(K)$ denotes the set of degree $d$ rational
functions $f\in K(x)$, and we write $\operatorname{Poly}_d^1(K)$ for the set of
degree $d$ polynomials $f\in K[x]$. We define
\begin{align*}
\operatorname{Poly}_d^1(\mathbb{F}_q)^\mathrm{stab} &:= \bigl\{ f\in\operatorname{Poly}_d^1(\math... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 48
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,426 | AMR-126-0049 | Open Problems in Arithmetic Dynamics — Conjecture 49 | Let $f(x)\in\mathbb{Z}[x]$ be a polynomial of degree at least $2$, and
let $\alpha\in\mathbb{Z}$ be a wandering point for $f$. Then $\mathcal{O}_f(\alpha)$
contains only finitely many primes. | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 49
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,427 | AMR-126-0050 | Open Problems in Arithmetic Dynamics — Conjecture 50 | Let $K$ be a number field, let $f(x)\in K(x)$ be a rational function
of degree at least $2$, let $\alpha\in K$ be a wandering point for $f$,
let $\beta\in K$, and let $c$ be a constant. Then the set
\[
\left\{ n\ge0 : \sum_{{\mathfrak{p}}} \operatorname{ord}_{\mathfrak{p}}^{\scriptscriptstyle+}\bigl(f^n(\alpha)-\beta\b... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 50
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,428 | AMR-126-0051 | Open Problems in Arithmetic Dynamics — Conjecture 51 | Let $f(x)\in\mathbb{Q}(x)$ be a rational function of degree at least $2$, and
let $\alpha,\beta\in\mathbb{Q}$ be $f$-wandering points.
Then the Zsigmondy set of ${\mathcal{A}}(f,\alpha,\beta)$ is finite. | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 51
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,429 | AMR-126-0052 | Open Problems in Arithmetic Dynamics — Conjecture 52 (Dynamical Lang--Siegel Conjecture for $\mathbb{P}^N$) | Let $f:\mathbb{P}^N\to\mathbb{P}^N$ be a morphism of degree $d\ge2$ defined over $\mathbb{Q}$,
and assume that for all $1\le n\le N+1$, the first coordinate function of $f^n$ is not a constant
multiple of $X_0^{d^n}$. Let $P\in\mathbb{P}^N(\mathbb{Q})$. Then $\mathcal{O}_f(P)$ contains
only finitely points that are int... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 52
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Dynamical Lang--Siegel Conjecture for $\... | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,430 | AMR-126-0053 | Open Problems in Arithmetic Dynamics — Question 53 | Let $f:\mathbb{P}^N\to\mathbb{P}^N$ be a morphism of degree $d$ defined over $\mathbb{Q}$.
\begin{parts}
(a)
For which $n\ge1$ is it true that $f$ has a point of exact
period $n$ in $\mathbb{P}^N(\mathbb{Q})$ if and only if $f$ has a point of exact
period $n$ in $\mathbb{P}^N(\mathbb{R})$ and in $\mathbb{P}^N(\mathbb{... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Question 53
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,431 | AMR-126-0054 | Open Problems in Arithmetic Dynamics — Conjecture 54 (Dynamical Brauer--Manin Conjecture) | \textup{(Hsia--Silverman, 2009, [MR2537714])}
Let $K$ be a number field, let $X/K$ be a variety, let $f:X\to X$ be
a morphism defined over $K$, let $Y\subset X$ be a subvariety
defined over $K$, and let $P\in X(K)$.
Let ${\mathcal{A}}_K$ be the ring of adeles of $K$, and for any set $S$,
let ${\mathcal{C}}(S)$ denote t... | Source list: Benedetto, DeMarco, Ingram, Jones et al. - Current Trends and Open Problems in Arithmetic Dynamics (2018)
Source item: Conjecture 54
Source URL: https://arxiv.org/abs/1806.04980
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: Dynamical Brauer--Manin Conjecture | 3 | open | Benedetto, DeMarco, Ingram, Jones et al. | 2,018 | 11 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,432 | AMR-127-0001 | Analysis of Boolean Functions — Correlation Bounds for Polynomials | Find an explicit (i.e., in $\mathsf{NP}$) function $f \colon \mathbb{F}_2^n \to \mathbb{F}_2$ such that we have the correlation bound $|\mathbb{E}[(-1)^{\langle f(\mathbf{x}), p(\mathbf{x}) \rangle }]| \leq 1/n$ for every $\mathbb{F}_2$-polynomial $p \colon \mathbb{F}_2^n \to \mathbb{F}_2$ of degree at most $\log_2 n$. | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 1 (Correlation Bounds for Polynomials)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: Folklore dating back to [Raz87,Smo87]
Remarks:
- Th... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,433 | AMR-127-0002 | Analysis of Boolean Functions — Tomaszewski's Conjecture | Let $a \in \mathbb{R}^n$ have $\|a\|_2 = 1$. Then $\Pr_{\mathbf{x} \sim \{-1,1\}^n}[|\langle a, \mathbf{x} \rangle | \leq 1] \geq 1/2$. | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 2 (Tomaszewski's Conjecture)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: Question attributed to Tomaszewski in [Guy89]
Remarks:
- The ... | 3 | solved | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Proposed/Source: Question attributed to Tomaszewski in [Guy89]\n\nRemarks:\n- The bound of $1/2$ would be sharp in light of $a = (1/\\sqrt{2}, 1/\\sqrt{2})$.\n\n- Holman and Kleitman [HK92] proved the lower bound $3/8... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,434 | AMR-127-0003 | Analysis of Boolean Functions — Talagrand's ``Convolution with a Biased Coin'' Conjecture | Let $f \colon \{-1,1\}^n \to \mathbb{R}^{\geq 0}$ have $\mathbb{E}[f] = 1$. Fix any $0 < \rho < 1$. Then $\operatorname{Pr}[\mathbb{T}_\rho f \geq t] < o(1/t)$. | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 3 (Talagrand's ``Convolution with a Biased Coin'' Conjecture)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [Tal89]
Remarks:
- Talagrand... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,435 | AMR-127-0004 | Analysis of Boolean Functions — Gotsman--Linial Conjecture | Among degree-$k$ polynomial threshold functions $f \colon \{-1,1\}^n \to \{-1,1\}$, the one with maximal total influence is the symmetric one $f(x) = \operatorname{sgn}(p(x_1 + \cdots + x_n))$, where $p$ is a degree-$k$ univariate polynomial which alternates sign on the $k+1$ values of $x_1 + \cdots + x_n$ closest to $... | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 5 (Gotsman--Linial Conjecture)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [GL94]
Remarks:
- The case $k = 1$ is easy.
- Slightly wea... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,436 | AMR-127-0005 | Analysis of Boolean Functions — Polynomial Freiman--Ruzsa Conjecture (in the $\mathbb{F}_2^n$ setting) | Suppose $\emptyset \neq A \subseteq \mathbb{F}_2^n$ satisfies $|A + A| \leq C|A|$. Then $A$ can be covered by the union of $\operatorname{poly}(C)$ affine subspaces, each of cardinality at most $|A|$. | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 6 (Polynomial Freiman--Ruzsa Conjecture (in the $\mathbb{F}_2^n$ setting))
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: Attributed to Ma... | 3 | solved | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Proposed/Source: Attributed to Marton in [Ruz93]; for the $\\mathbb{F}_2^n$ version, see e.g. [Gre05]\n\nRemarks:\n- The following conjecture is known to be equivalent: Suppose $f \\colon \\mathbb{F}_2^n \\to \\mathbb... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,437 | AMR-127-0006 | Analysis of Boolean Functions — Mansour's Conjecture | Let $f \colon \{-1,1\}^n \to \{-1,1\}$ be computable by a DNF of size $s > 1$ and let $\varepsilon \in (0,1/2]$. Then $f$'s Fourier spectrum is $\varepsilon$-concentrated on a collection $\mathcal{F}$ with $|\mathcal{F}| \leq s^{O(\log(1/\varepsilon))}$. | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 7 (Mansour's Conjecture)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [Man94]
Remarks:
- Weaker version: replacing $s^{O(\log(1/\vareps... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,438 | AMR-127-0007 | Analysis of Boolean Functions — Bernoulli Conjecture | Let $T$ be a finite collection of vectors in $\mathbb{R}^n$. Define $b(T) = \mathbb{E}_{\mathbf{x} \sim \{-1,1\}^n}[\max_{t \in T} \langle t, \mathbf{x} \rangle ]$, and define $g(T)$ to be the same quantity except with $\mathbf{x} \sim \mathbb{R}^n$ Gaussian. Then there exists a finite collection of vectors $T'$ such t... | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 8 (Bernoulli Conjecture)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [Tal94a]
Remarks:
- The quantity $g(T)$ is well-understood in ter... | 3 | solved | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Proposed/Source: [Tal94a]\n\nRemarks:\n- The quantity $g(T)$ is well-understood in terms of the geometry of $T$, thanks to Talagrand's majorizing measures theorem.\n\n- Talagrand offers a \\$5000 prize for proving thi... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,439 | AMR-127-0008 | Analysis of Boolean Functions — Fourier Entropy--Influence Conjecture | There is a universal constant $C$ such that for any $f \colon \{-1,1\}^n \to \{-1,1\}$ it holds that $\mathbb{H}[\widehat{f}^2] \leq C \cdot \mathbf{I}[f]$, where $\mathbb{H}[\widehat{f}^2] = \sum_{S} \widehat{f}(S)^2 \log_2\frac{1}{\widehat{f}(S)^2}$ is the spectral entropy and $\mathbf{I}[f]$ is the total influence. | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 9 (Fourier Entropy--Influence Conjecture)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [FK96b]
Remarks:
- Proved for ``almost all'' pol... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,440 | AMR-127-0009 | Analysis of Boolean Functions — Majority Is Least Stable Conjecture | Let $f \colon \{-1,1\}^n \to \{-1,1\}$ be a linear threshold function, $n$ odd. Then for all $\rho \in [0,1]$, $\Stab_\rho[f] \geq \Stab_\rho[\maj_n]$. | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 10 (Majority Is Least Stable Conjecture)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [BKS99]
Remarks:
- Slightly weaker version: If $f... | 3 | solved | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Proposed/Source: [BKS99]\n\nRemarks:\n- Slightly weaker version: If $f$ is a linear threshold function then $\\NS_\\delta[f] \\leq \\frac{2}{\\pi} \\sqrt{\\delta} + o(\\sqrt{\\delta})$.\n\n- The best result towards th... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,441 | AMR-127-0010 | Analysis of Boolean Functions — Optimality of Majorities for Non-Interactive Correlation Distillation | Fix $r \in \mathbb{N}$, $n$ odd, and $0 < \varepsilon < 1/2$. For $f \colon \{-1,1\}^n \to \{-1,1\}$, define $P(f) = \operatorname{Pr}[f(\mathbf{y}^{(1)}) = f(\mathbf{y}^{(2)}) = \cdots f(\mathbf{y}^{(r)})]$, where $\mathbf{x} \sim \{-1,1\}^n$ is chosen uniformly and then each $\mathbf{y}^{(i)}$ is (independently) an $... | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 11 (Optimality of Majorities for Non-Interactive Correlation Distillation)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [MO05] (original... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,442 | AMR-127-0011 | Analysis of Boolean Functions — Noise Sensitivity of Intersections of Halfspaces | Let $f \colon \{-1,1\}^n \to \{-1,1\}$ be the intersection (AND) of $k$ linear threshold functions. Then $\NS_\delta[f] \leq O(\sqrt{\log k}) \cdot \sqrt{\delta}$. | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 12 (Noise Sensitivity of Intersections of Halfspaces)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [KOS02]
Remarks:
- The bound $O(k) \... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,443 | AMR-127-0012 | Analysis of Boolean Functions — Non-Interactive Correlation Distillation with Erasures | Let $f \colon \{-1,1\}^n \to \{-1,1\}$ be an unbiased function. Let $\mathbf{z} \sim \{-1,0,1\}^n$ be a ``random restriction'' in which each coordinate $\bz_i$ is (independently) $\pm 1$ with probability $p/2$ each, and $0$ with probability $1-p$. Assuming $p < 1/2$ and $n$ odd, is it true that $\mathbb{E}_{\mathbf{z}}... | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 13 (Non-Interactive Correlation Distillation with Erasures)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [Yan04]
Remarks:
- For $p \geq... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,444 | AMR-127-0013 | Analysis of Boolean Functions — Triangle Removal in $\mathbb{F}_2^n$ | Let $A \subseteq \mathbb{F}_2^n$. Suppose that $\varepsilon 2^n$ elements must be removed from $A$ in order to make it ``triangle-free'' (meaning there does not exist $x, y, x+y \in A$). Is it true that $\Pr_{\mathbf{x}, \mathbf{y}}[\mathbf{x}, \mathbf{y}, \mathbf{x}+\mathbf{y} \in A] \geq \operatorname{poly}(\varepsil... | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 14 (Triangle Removal in $\mathbb{F}_2^n$)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [Gre05a]
Remarks:
- Green [Gre05a] showed the lo... | 3 | solved | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Proposed/Source: [Gre05a]\n\nRemarks:\n- Green [Gre05a] showed the lower bound $1/(2\\!\\uparrow\\!\\uparrow\\!\\varepsilon^{-\\Theta(1)})$.\n\n- Bhattacharyya and Xie [BX10] constructed an $A$ for which the probabili... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,445 | AMR-127-0014 | Analysis of Boolean Functions — Subspaces in Sumsets | Fix a constant $\alpha > 0$. Let $A \subseteq \mathbb{F}_2^n$ have density at least $\alpha$. Is it true that $A+A$ contains a subspace of codimension $O(\sqrt{n})$? | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 15 (Subspaces in Sumsets)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [Gre05a]
Remarks:
- The analogous problem for the group $Z_N$ da... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,446 | AMR-127-0015 | Analysis of Boolean Functions — Aaronson--Ambainis Conjecture | Let $f \colon \{-1,1\}^n \to [-1,1]$ have degree at most $k$. Then there exists $i \in [n]$ with $\Inf_i[f] \geq (\operatorname{Var}[f]/k)^{O(1)}$. | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 16 (Aaronson--Ambainis Conjecture)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [Aar08,AA11]
Remarks:
- True for $f \colon \{-1,1\}^n \... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,447 | AMR-127-0016 | Analysis of Boolean Functions — Bhattacharyya--Grigorescu--Shapira Conjecture | Let $M \in \mathbb{F}_2^{m \times k}$ and $\sigma \in \{0,1\}^k$. Say that $f \colon \mathbb{F}_2^n \to \{0,1\}$ is *$(M,\sigma)$-free* if there does not exist $X = (x^{(1)}, \dots, x^{(k)})$ (where each $x^{(j)} \in \mathbb{F}_2^n$ is a row vector) such that $MX = 0$ and $f(x^{(j)}) = \sigma_j$ for all $j \in [k]$. No... | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 17 (Bhattacharyya--Grigorescu--Shapira Conjecture)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [BGS10]
Remarks:
- The conjecture is mo... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,448 | AMR-127-0017 | Analysis of Boolean Functions — Symmetric Gaussian Problem | Fix $0 \leq \rho, \mu, \nu \leq 1$. Suppose $A,B \subseteq \mathbb{R}^n$ have Gaussian measure $\mu$, $\nu$ respectively. Further, suppose $A$ is centrally symmetric: $A = -A$. What is the minimal possible value of $\operatorname{Pr}[\mathbf{x} \in A, \mathbf{y} \in B]$, when $(\mathbf{x}, \mathbf{y})$ are $\rho$-corre... | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 18 (Symmetric Gaussian Problem)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [CR10]
Remarks:
- It is equivalent to require both $A = -A... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,449 | AMR-127-0018 | Analysis of Boolean Functions — Standard Simplex Conjecture | Fix $0 \leq \rho \leq 1$. Then among all partitions of $\mathbb{R}^n$ into $3 \leq q \leq n+1$ parts of equal Gaussian measure, the maximal noise stability at $\rho$ occurs for a ``standard simplex partition''. By this it is meant a partition $A_1, \dots, A_q$ satisfying $A_i \supseteq \{x \in \mathbb{R}^n : \langle a_... | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 19 (Standard Simplex Conjecture)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [IM09]
Remarks:
- Implies the Plurality Is Stablest Conje... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,450 | AMR-127-0019 | Analysis of Boolean Functions — $k$-wise Independence for PTFs | Fix $d \in \mathbb{N}$ and $\varepsilon \in (0,1)$. Determine the least $k = k(d,\varepsilon)$ such that the following holds: If $p \colon \mathbb{R}^n \to \mathbb{R}$ is any degree-$d$ multivariate polynomial, and $\mathbf{X}$ is any $\mathbb{R}^n$-valued random variable with the property that each $\bX_i$ has the sta... | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 21 ($k$-wise Independence for PTFs)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [DGJ+09]
Remarks:
- For $d = 1$, Diakonikolas, Gopalan... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,451 | AMR-127-0020 | Analysis of Boolean Functions — $\varepsilon$-biased Sets for DNFs | Is it true for each constant $\delta > 0$ that $s^{-O(1)}$-biased densities $\delta$-fool size-$s$ DNFs? I.e., that if $f \colon \{0,1\}^n \to \{-1,1\}$ is computable by a size-$s$ DNF and $\varphi$ is an $s^{-O(1)}$-biased density on $\{0,1\}$, then $|\mathbb{E}_{\mathbf{x} \sim \{0,1\}^n}[f(\mathbf{x})] - \mathbb{E}_... | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 22 ($\varepsilon$-biased Sets for DNFs)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: [DETT10], though the problem of pseudorandom genera... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,452 | AMR-127-0021 | Analysis of Boolean Functions — PTF Sparsity for Inner Product Mod 2 | Is it true that any PTF representation of the inner product mod $2$ function on $2n$ bits, $\IP_{2n} \colon \mathbb{F}_2^{2n} \to \{-1,1\}$, requires at least $3^n$ monomials? | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 23 (PTF Sparsity for Inner Product Mod 2)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: Srikanth Srinivasan, 2010
Remarks:
- Rocco Serve... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,453 | AMR-127-0022 | Analysis of Boolean Functions — Servedio--Tan--Verbin Conjecture (Disproved) | Fix any $\varepsilon > 0$. Then every monotone $f \colon \{-1,1\}^n \to \{-1,1\}$ is $\varepsilon$-close to a $\operatorname{poly}(\deg(f))$-junta. | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 24 (Servedio--Tan--Verbin Conjecture (Disproved))
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: Elad Verbin (2010) and independently Rocc... | 3 | solved | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | {
"evidence": "published",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Proposed/Source: Elad Verbin (2010) and independently Rocco Servedio and Li-Yang Tan (2010)\n\nRemarks:\n- One can equivalently replace degree by decision-tree depth or maximum sensitivity.\n\n- RESOLVED (in the negat... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,454 | AMR-127-0023 | Analysis of Boolean Functions — Average versus Max Sensitivity for Monotone Functions | Let $f \colon \{-1,1\}^n \to \{-1,1\}$ be monotone. Then $\mathbf{I}[f] < o(\operatorname{sens}[f])$. | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 25 (Average versus Max Sensitivity for Monotone Functions)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: Rocco Servedio, Li-Yang Tan, 201... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,455 | AMR-127-0024 | Analysis of Boolean Functions — Approximate Degree for Approximate Majority | What is the least possible degree of a function $f \colon \{-1,1\}^n \to [-1, -2/3] \cup [2/3, 1]$ which has $f(x) \in [2/3,1]$ whenever $\sum_{i=1}^n x_i \geq n/2$ and has $f(x) \in [-1,-2/3]$ whenever $\sum_{i=1}^n x_i \leq -n/2$? | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 26 (Approximate Degree for Approximate Majority)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: Srikanth Srinivasan, 2010
Remarks:
- Note... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,456 | AMR-127-0025 | Analysis of Boolean Functions — Uncertainty Principle for Quadratic Fourier Analysis | Suppose $q_1, \dots, q_m \colon \mathbb{F}_2^n \to \mathbb{F}_2$ are polynomials of degree at most $2$ and suppose the indicator function of $(1, \dots, 1) \in \mathbb{F}_2^n$, namely $\operatorname{AND} \colon \mathbb{F}_2^n \to \{-1,1\}$, is expressible as $\operatorname{AND}(x) = \sum_{i=1}^m c_i (-1)^{q_i(x)}$ for ... | Source list: O'Donnell - Open Problems in Analysis of Boolean Functions (2012)
Source item: Problem 27 (Uncertainty Principle for Quadratic Fourier Analysis)
Source URL: https://arxiv.org/abs/1204.6447
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Proposed/Source: Hamed Hatami, 2011
Remarks:
- Ha... | 3 | open | O'Donnell | 2,012 | 15 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,457 | AMR-128-0001 | Spaces with Lower Ricci Curvature — Conjecture 1 | Let $\big(M^n_i,g_i,p_i\big)\to (X,d,p)$ satisfy $\mathrm{Ric}_i\geq -\lambda$ and $\mathrm{Vol}(B_1(p_i))>v>0$, then $X$ is homeomorphic to a manifold away from a set of codimension three. | Source list: Naber - Conjectures and Open Questions on Spaces with Lower Ricci Curvature (2020)
Source item: Conjecture 1
Source URL: https://arxiv.org/abs/2010.10031
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Naber | 2,020 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,458 | AMR-128-0002 | Spaces with Lower Ricci Curvature — Conjecture 2 | Let $\big(M^n_i,g_i,p_i\big)\to (X,d,p)$ satisfy $\mathrm{Ric}\geq -(n-1)$ and $\mathrm{Vol}(B_1(p_i))>v>0$, then the tangent cones of $X$ are unique away from a set of codimension three. | Source list: Naber - Conjectures and Open Questions on Spaces with Lower Ricci Curvature (2020)
Source item: Conjecture 2
Source URL: https://arxiv.org/abs/2010.10031
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Naber | 2,020 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,459 | AMR-128-0003 | Spaces with Lower Ricci Curvature — Conjecture 3 | Let $\big(M^n_i,g_i,p_i\big)\to (X,d,p)$ satisfy $\mathrm{Ric}\geq -(n-1)$ and $\mathrm{Vol}(B_1(p_i))>v>0$, then the tangent cones of $X$ are all homoemorphic at a fixed point away from a set of codimension five. | Source list: Naber - Conjectures and Open Questions on Spaces with Lower Ricci Curvature (2020)
Source item: Conjecture 3
Source URL: https://arxiv.org/abs/2010.10031
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Naber | 2,020 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,460 | AMR-128-0004 | Spaces with Lower Ricci Curvature — Conjecture 4 | Let $\big(M^n_i,g_i,p_i\big)\to (X,d,p)$ satisfy $\mathrm{Ric} \geq -\lambda$ and $\mathrm{Vol}(B_1(p_i))>v>0$, then $\mathcal{S}^k$ is contained in a union of $k$-dimensional submanifolds. More precisely, there exist a countable collection of bi-H\"older maps $\varphi_i\colon B_1\big(0^k\big)\to M$ such that if $\hat ... | Source list: Naber - Conjectures and Open Questions on Spaces with Lower Ricci Curvature (2020)
Source item: Conjecture 4
Source URL: https://arxiv.org/abs/2010.10031
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Naber | 2,020 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,461 | AMR-128-0005 | Spaces with Lower Ricci Curvature — Conjecture 5 | Let $\big(M^n_i,g_i,p_i\big)\to (X,d,p)$ satisfy $\mathrm{Ric}\geq -\lambda$ and $\mathrm{Vol}(B_1(p_i))>v>0$, then for each $k\leq n-2$ there exists $\hat\mathcal{S}^k$ with $\dim\hat\mathcal{S}^k\leq k$ such that for each $x\in X\setminus \hat\mathcal{S}^k$ we have that *every* tangent cone is at least $k+1$-symmetri... | Source list: Naber - Conjectures and Open Questions on Spaces with Lower Ricci Curvature (2020)
Source item: Conjecture 5
Source URL: https://arxiv.org/abs/2010.10031
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Naber | 2,020 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,462 | AMR-128-0006 | Spaces with Lower Ricci Curvature — Conjecture 6 (noncollapsing Yau's conjecture) | Let $\big(M^n,g,x\big)$ satisfy
\[ \mathrm{Ric}\geq -\lambda \qquad \text{and} \qquad \mathrm{Vol}(B_1(p))>v>0,\] then $\exists$ $C(n,\lambda,v)>0$ such that
\begin{gather*}
\fint_{B_1(x)} R < C .
\end{gather*} | Source list: Naber - Conjectures and Open Questions on Spaces with Lower Ricci Curvature (2020)
Source item: Conjecture 6
Source URL: https://arxiv.org/abs/2010.10031
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: noncollapsing Yau's conjecture | 3 | open | Naber | 2,020 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,463 | AMR-128-0007 | Spaces with Lower Ricci Curvature — Conjecture 7 (energy identity) | Let $(M^n_j,g_j,x_j)\to (X,d,x)$ satisfy $\mathrm{Ric}_j\geq -\lambda$ and the noncollapsing condition $\mathrm{Vol}(B_1(x_j))\geq v>0$, then as measures the scalar curvature converges
\begin{gather*}
R_j \mathrm{d}v_j\to R_X \mathcal{H}^n+2\theta_x \mathcal{H}^{n-1}_\Sigma+2\pi(1-r_x)\mathcal{H}^{n-2}_S ,
\end{gather*... | Source list: Naber - Conjectures and Open Questions on Spaces with Lower Ricci Curvature (2020)
Source item: Conjecture 7
Source URL: https://arxiv.org/abs/2010.10031
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: energy identity | 3 | open | Naber | 2,020 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,464 | AMR-128-0008 | Spaces with Lower Ricci Curvature — Conjecture 8 | Let $\gamma\colon (-2,2)\to X$ be a minimizing geodesic, then for $s,t\in (-1,1)$ we have that
\itemsep=0pt
$1)$ $($Lipschitz rate$)$ $\frac{\nu(B_r(\gamma(s)))}{\nu(B_r(\gamma(t)))}\leq C(n,\lambda)|t-s|$,
$2)$ $($constant density$)$ if $X$ is noncollapsed, then $\lim\limits_{r\to 0}\frac{\nu(B_r(\gamma(s)))}{\nu(B... | Source list: Naber - Conjectures and Open Questions on Spaces with Lower Ricci Curvature (2020)
Source item: Conjecture 8
Source URL: https://arxiv.org/abs/2010.10031
Accessed: 2026-10-16
Extraction: arxiv_latex | 3 | open | Naber | 2,020 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,465 | AMR-128-0009 | Spaces with Lower Ricci Curvature — Conjecture 9 (local Yau conjecture) | Finally, let us end by discussing the regularity of manifolds with lower Ricci bounds, potentially including those with small volume. The main conjecture out there is one due to Yau, and we state a local version of it below:
Let $\big(M^n,g\big)$ satisfy $\mathrm{Ric}\geq -\lambda$, then $\int_{B_1(p)} R \leq C(n,\lam... | Source list: Naber - Conjectures and Open Questions on Spaces with Lower Ricci Curvature (2020)
Source item: Conjecture 9
Source URL: https://arxiv.org/abs/2010.10031
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Attribution/Context: local Yau conjecture | 3 | open | Naber | 2,020 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,466 | AMR-129-0001 | Billiards and Quantitative Symplectic Geometry — Misha Bialy — Problem 1 | **Magnetic billiards for strong magnetic fields..**
Magnetic billiards were introduced and studied in [berry-robnik]. In this model, the billiard ball moves under the influence of a constant magnetic field along Larmor circles (for a magnetic field of magnitude $B$, the Larmor circles have radius $1/B$). The only know... | Source list: Albach, Barraud et al. - Open Problems in Billiards and Quantitative Symplectic Geometry (2026)
Source item: Misha Bialy — Problem 1
Source URL: https://arxiv.org/abs/2602.12896
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Contributor: Misha Bialy | 3 | open | Albach, Barraud et al. | 2,026 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,467 | AMR-129-0002 | Billiards and Quantitative Symplectic Geometry — Misha Bialy — Problem 2 | **Minkowski billiard..**
The model of Minkowski and, more generally, Finsler billiards, was introduced and studied by Gutkin and Tabachnikov in [GUTKIN2002277]. It plays an important role in a recent study by Artstein-Avidan, Karasev, and Ostrover [Artstein_Avidan_2014] on Mahler's conjecture. Consider a smooth convex... | Source list: Albach, Barraud et al. - Open Problems in Billiards and Quantitative Symplectic Geometry (2026)
Source item: Misha Bialy — Problem 2
Source URL: https://arxiv.org/abs/2602.12896
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Contributor: Misha Bialy | 3 | open | Albach, Barraud et al. | 2,026 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,468 | AMR-129-0003 | Billiards and Quantitative Symplectic Geometry — Misha Bialy — Problem 3 | **Newton equations with $1\frac 12$ degrees of freedom, with periodic potential..**
Newton's equation $\ddot u=-V'_u(u,t)$ is the Euler-Lagrange equation associated with the Lagrangian $L(u,\dot u, t)=\frac 12\dot u^2-V(u,t)$. Assume that $V$ is periodic both in $u$ and in $t$. The corresponding Hamiltonian flow acts ... | Source list: Albach, Barraud et al. - Open Problems in Billiards and Quantitative Symplectic Geometry (2026)
Source item: Misha Bialy — Problem 3
Source URL: https://arxiv.org/abs/2602.12896
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Contributor: Misha Bialy | 3 | open | Albach, Barraud et al. | 2,026 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
800,469 | AMR-129-0004 | Billiards and Quantitative Symplectic Geometry — Misha Bialy — Problem 4 | **An elliptic PDE.**
Consider an elliptic PDE $\Delta u=-V'_u(u, x_1, \ldots ,x_n)$, where $\Delta=\sum_{i=1}^{n}\partial^2_{x_i}$ and $V$ is a 1-periodic function of $u, x_1,\ldots ,x_n$. This equation is the Euler-Lagrange equation for the functional $\int \left( \frac 12|\nabla u|^2-V(u, x_1, \ldots, x_n)\right) dx... | Source list: Albach, Barraud et al. - Open Problems in Billiards and Quantitative Symplectic Geometry (2026)
Source item: Misha Bialy — Problem 4
Source URL: https://arxiv.org/abs/2602.12896
Accessed: 2026-10-16
Extraction: arxiv_latex
Notes and literature:
Contributor: Misha Bialy | 3 | open | Albach, Barraud et al. | 2,026 | 6 | 13 | 0 | 0 | 2026-10-16T14:00:00 | 2026-10-16T14:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 13,
"name": "amr_open_problem_lists",
"display_name": "AMR Open Problem Lists",
"description": "Open and partially solved problems collected from public source lists indexed by AMR, with per-record provenance and status notes.",
"slug": "amr-open-problem-lists",
"order_index": 13,
"created_at": "2... | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.