| import OAI.NumberTheory.Jacobsthal.Harmonic.PolynomialLocalGraph |
| import OAI.NumberTheory.Jacobsthal.Partitions.InflectionTransport |
| import OAI.NumberTheory.Jacobsthal.Paths.AffineBranch |
| |
| namespace OAI |
|
|
| namespace Erdos970 |
|
|
| section |
|
|
| namespace ErdosDivisibleInflection |
|
|
| open ErdosCriticalGeometry ErdosImplicitCurvature |
|
|
| noncomputable def swapVars (R : Type*) [CommSemiring R] : MV R ≃ₐ[R] MV R := |
| MvPolynomial.renameEquiv R (Equiv.swap (0 : Fin 2) 1) |
|
|
| theorem swap_twice (R : Type*) [CommSemiring R] (Q : MV R) : |
| swapVars R (swapVars R Q) = Q := by |
| simpa [swapVars] using (MvPolynomial.renameEquiv R (Equiv.swap (0 : Fin 2) 1)).apply_symm_apply Q |
|
|
| theorem nestedX_swap (R : Type*) [CommSemiring R] (Q : MV R) : |
| nestedX R (swapVars R Q) = nestedY R Q := rfl |
|
|
| theorem nestedY_swap (R : Type*) [CommSemiring R] (Q : MV R) : |
| nestedY R (swapVars R Q) = nestedX R Q := by |
| change nestedX R (swapVars R (swapVars R Q)) = _ |
| rw [swap_twice] |
| |
| theorem partial_zero_swap (R : Type*) [CommSemiring R] (Q : MV R) : |
| MvPolynomial.pderiv 0 (swapVars R Q) = swapVars R (MvPolynomial.pderiv 1 Q) := by |
| apply (nestedX R).injective |
| rw [nestedX_partial_zero,nestedX_swap,nestedX_swap,nestedY_partial_one] |
|
|
| theorem partial_one_swap (R : Type*) [CommSemiring R] (Q : MV R) : |
| MvPolynomial.pderiv 1 (swapVars R Q) = swapVars R (MvPolynomial.pderiv 0 Q) := by |
| apply (nestedY R).injective |
| rw [nestedY_partial_one,nestedY_swap,nestedY_swap,nestedX_partial_zero] |
|
|
| theorem eval_swap (R : Type*) [CommSemiring R] (Q : MV R) (x y : R) : |
| MvPolynomial.eval ![x,y] (swapVars R Q) = MvPolynomial.eval ![y,x] Q := by |
| have h1 := eval_nestedX R (swapVars R Q) x y |
| rw [nestedX_swap] at h1 |
| exact h1.symm.trans (eval_nestedY R Q y x) |
|
|
| theorem peval_swap (Q : MV ℝ) (x y : ℝ) : peval (swapVars ℝ Q) x y = peval Q y x := by |
| rw [peval_eq_eval_pair,peval_eq_eval_pair,eval_swap] |
|
|
| theorem complexify_swap (Q : MV ℝ) : complexify (swapVars ℝ Q) = swapVars ℂ (complexify Q) := by |
| simp only [complexify,swapVars,MvPolynomial.renameEquiv_apply,MvPolynomial.map_rename] |
| |
| theorem mixed_partials_complex (Q : MV ℂ) : |
| MvPolynomial.pderiv 1 (MvPolynomial.pderiv 0 Q) = |
| MvPolynomial.pderiv 0 (MvPolynomial.pderiv 1 Q) := by |
| classical |
| ext m |
| simp [MvPolynomial.coeff_pderiv,add_assoc,add_left_comm,add_comm,mul_left_comm,mul_comm] |
| |
| theorem inflection_swap (Q : MV ℂ) : inflection (swapVars ℂ Q) = swapVars ℂ (inflection Q) := by |
| unfold inflection |
| simp only [partial_zero_swap,partial_one_swap,mixed_partials_complex,map_add,map_mul,map_pow] |
| have hC : swapVars ℂ (MvPolynomial.C (-2 : ℂ)) = MvPolynomial.C (-2 : ℂ) := by simp [swapVars] |
| rw [hC] |
| ring |
|
|
| theorem complex_divisibility_swap (Q : MV ℝ) |
| (hdiv : complexify Q ∣ inflection (complexify Q)) : |
| complexify (swapVars ℝ Q) ∣ inflection (complexify (swapVars ℝ Q)) := by |
| rw [complexify_swap,inflection_swap] |
| exact map_dvd (swapVars ℂ).toRingHom hdiv |
|
|
| end ErdosDivisibleInflection |
|
|
| end |
|
|
| section |
|
|
| namespace ErdosDivisibleInflection |
|
|
| open ErdosCriticalGeometry ErdosImplicitCurvature |
|
|
| theorem no_regular_y_point (Q : MV ℝ) (hQ : Irreducible (complexify Q)) |
| (hdeg : 1 < Q.totalDegree) (hdiv : complexify Q ∣ inflection (complexify Q)) |
| (x y : ℝ) (hzero : peval Q x y = 0) (hy : peval (partialY Q) x y ≠ 0) : False := by |
| obtain ⟨a,b,f,hax,hxb,hfx,hf,hcurve,hfy⟩ := exists_local_real_graph Q x y hzero hy |
| have hsecond := zero_second_derivative_on_divisible_graph Q f a b hdiv hf hcurve hfy |
| obtain ⟨A,B,haffine⟩ := affine_of_second_derivative_zero f a b x ⟨hax,hxb⟩ hf hsecond |
| have hdegC : 1 < (complexify Q).totalDegree := by rwa [complexify_totalDegree] |
| apply no_affine_interval (complexify Q) hQ hdegC (A : ℂ) (B : ℂ) a b (hax.trans hxb) |
| intro t ht |
| have hz := hcurve t ht |
| rw [haffine t ht] at hz |
| have hc := complexify_eval Q t (A*t+B) |
| rw [← peval_eq_eval_pair,hz] at hc |
| simpa only [Complex.ofReal_mul,Complex.ofReal_add,Complex.ofReal_zero] using hc |
|
|
| theorem no_regular_x_point (Q : MV ℝ) (hQ : Irreducible (complexify Q)) |
| (hdeg : 1 < Q.totalDegree) (hdiv : complexify Q ∣ inflection (complexify Q)) |
| (x y : ℝ) (hzero : peval Q x y = 0) (hx : peval (partialX Q) x y ≠ 0) : False := by |
| have hirr : Irreducible (complexify (swapVars ℝ Q)) := by |
| rw [complexify_swap] |
| exact (MulEquiv.irreducible_iff (swapVars ℂ).toMulEquiv).mpr hQ |
| have hd : 1 < (swapVars ℝ Q).totalDegree := by |
| simpa only [swapVars,MvPolynomial.totalDegree_renameEquiv] using hdeg |
| have hz : peval (swapVars ℝ Q) y x = 0 := by rw [peval_swap] |
| have hpart : peval (partialY (swapVars ℝ Q)) y x ≠ 0 := by |
| change peval (MvPolynomial.pderiv 1 (swapVars ℝ Q)) y x ≠ 0 |
| rw [partial_one_swap,peval_swap] |
| exact hx |
| exact no_regular_y_point (swapVars ℝ Q) hirr hd (complex_divisibility_swap Q hdiv) y x hz hpart |
|
|
| theorem no_real_regular_point_of_divisible_inflection (Q : MV ℝ) |
| (hQ : Irreducible (complexify Q)) (hdeg : 1 < Q.totalDegree) |
| (hdiv : complexify Q ∣ inflection (complexify Q)) (x y : ℝ) |
| (hzero : peval Q x y = 0) |
| (hregular : peval (partialX Q) x y ≠ 0 ∨ peval (partialY Q) x y ≠ 0) : False := by |
| rcases hregular with hx | hy |
| · exact no_regular_x_point Q hQ hdeg hdiv x y hzero hx |
| · exact no_regular_y_point Q hQ hdeg hdiv x y hzero hy |
| |
| theorem divisible_inflection_gradient_zero (Q : MV ℝ) |
| (hQ : Irreducible (complexify Q)) (hdeg : 1 < Q.totalDegree) |
| (hdiv : complexify Q ∣ inflection (complexify Q)) (x y : ℝ) |
| (hzero : peval Q x y = 0) : |
| peval (partialX Q) x y = 0 ∧ peval (partialY Q) x y = 0 := by |
| constructor |
| · by_contra hx |
| exact no_regular_x_point Q hQ hdeg hdiv x y hzero hx |
| · by_contra hy |
| exact no_regular_y_point Q hQ hdeg hdiv x y hzero hy |
| |
| theorem real_divisible_inflection_gradient_zero (Q : MV ℝ) |
| (hQ : Irreducible (complexify Q)) (hdeg : 1 < Q.totalDegree) |
| (hdiv : Q ∣ inflectionPolynomial Q) (x y : ℝ) (hzero : peval Q x y = 0) : |
| peval (partialX Q) x y = 0 ∧ peval (partialY Q) x y = 0 := |
| divisible_inflection_gradient_zero Q hQ hdeg (real_divisibility_to_complex Q hdiv) x y hzero |
| |
| end ErdosDivisibleInflection |
| |
| end |
| |
| end Erdos970 |
| |
| end OAI |
| |
