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mathematics
inverse-problems
observability
noncommutative-algebra
symbolic-computation
calibration-rigidity
| import sympy as sp | |
| import pytest | |
| from noulith import * | |
| def test_penultimate_ideal_universality(m): | |
| a,b=sp.symbols('a b') | |
| generators=[a*a,b*b,a*b,a**3+b**4,a*a*b,b**5][:m*(m-1)//2] | |
| alg=universal_algebra(generators,m=m) | |
| sh=alg.shuffle_table() | |
| full=(1<<m)-1 | |
| actual=ideal_generators(sh,full,m-1) | |
| assert equal_polynomial_ideals(actual,generators,[a,b]) | |
| for mask in range(full): | |
| assert 1 in ideal_generators(sh,mask,m-1) | |
| def test_arc_blind_pair_exact_ideals(): | |
| a,b=sp.symbols('a b') | |
| left=universal_algebra([a*a,b*b,0],m=3) | |
| right=universal_algebra([a*a,b*b,a*b],m=3) | |
| il=ideal_generators(left.shuffle_table(),7,2) | |
| ir=ideal_generators(right.shuffle_table(),7,2) | |
| assert equal_polynomial_ideals(il,[a*a,b*b],[a,b]) | |
| assert equal_polynomial_ideals(ir,[a*a,a*b,b*b],[a,b]) | |
| assert not equal_polynomial_ideals(il,ir,[a,b]) | |
| assert sp.groebner(il,a,b).reduce(a*b)[1]==a*b | |
| assert sp.groebner(ir,a,b).reduce(a*b)[1]==0 | |
| def test_arbitrarily_large_arc_blind_defect_gap(p): | |
| j=[(p,0),(0,p)] | |
| k=[(i,p-i) for i in range(p+1)] | |
| sj=standard_monomials(j,2*p) | |
| sk=standard_monomials(k,2*p) | |
| assert len(sj)==p*p | |
| assert len(sk)==p*(p+1)//2 | |
| assert len(sj)-len(sk)==p*(p-1)//2 | |
| def test_arc_valuation_equality_monomial_oracle(order_a,order_b): | |
| for p in range(2,10): | |
| lhs=min(p*order_a,p*order_b) | |
| rhs=min(i*order_a+(p-i)*order_b for i in range(p+1)) | |
| assert lhs==rhs | |
| def test_nonlinear_arc_substitution_with_cancellation_inside_coordinates(): | |
| a,b,t=sp.symbols('a b t') | |
| arcs=[(t**2+t**4,t**3-t**5),(t**4,t**4+t**7),(0,t**5),(t**3,0),(0,0)] | |
| for at,bt in arcs: | |
| left=min(valuation(at**2,t),valuation(bt**2,t)) | |
| right=min(valuation(at**2,t),valuation(at*bt,t),valuation(bt**2,t)) | |
| assert left==right | |
| def test_hilbert_functions_and_generator_minimality(): | |
| sj=standard_monomials([(2,0),(0,2)],5) | |
| sk=standard_monomials([(2,0),(1,1),(0,2)],5) | |
| assert [sum(sum(x)==k for x in sj) for k in range(4)]==[1,2,1,0] | |
| assert [sum(sum(x)==k for x in sk) for k in range(4)]==[1,2,0,0] | |
| assert len(sj)==4 and len(sk)==3 | |
| def test_exact_first_and_second_betti_certificates(): | |
| a,b=sp.symbols('a b') | |
| row=sp.Matrix([[a*a,a*b,b*b]]) | |
| syzygy=sp.Matrix([[-b,0],[a,-b],[0,a]]) | |
| assert row*syzygy==sp.zeros(1,2) | |
| assert [sp.expand(syzygy.extract(rows,[0,1]).det()) for rows in [[1,2],[0,2],[0,1]]]==[a*a,-a*b,b*b] | |