def test_Submodule_represent(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) C = B.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ)) a0 = A(to_col([6, 12, 18, 24])) a1 = A(to_col([2, 4, 6, 8])) a2 = A(to_col([1, 3, 5, 7])) b1 = B.represent(a1) assert b1.flat() == [1, 2, 3, 4] c0 = C.represent(a0) assert c0.flat() == [1, 2, 3, 4] Y = A.submodule_from_matrix( DomainMatrix([ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], ], (3, 4), ZZ).transpose()) U = Poly(cyclotomic_poly(7, x)) Z = PowerBasis(U) z0 = Z(to_col([1, 2, 3, 4, 5, 6])) raises(ClosureFailure, lambda: Y.represent(A(3))) raises(ClosureFailure, lambda: B.represent(a2)) raises(ClosureFailure, lambda: B.represent(z0))
def test_Submodule_reduced(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) C = A.submodule_from_matrix(6 * DomainMatrix.eye(4, ZZ), denom=3) D = C.reduced() assert D.denom == 1 and D == C == B
def test_Submodule_reduce_element(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.whole_submodule() b = B(to_col([90, 84, 80, 75]), denom=120) C = B.submodule_from_matrix(DomainMatrix.eye(4, ZZ), denom=2) b_bar_expected = B(to_col([30, 24, 20, 15]), denom=120) b_bar = C.reduce_element(b) assert b_bar == b_bar_expected C = B.submodule_from_matrix(DomainMatrix.eye(4, ZZ), denom=4) b_bar_expected = B(to_col([0, 24, 20, 15]), denom=120) b_bar = C.reduce_element(b) assert b_bar == b_bar_expected C = B.submodule_from_matrix(DomainMatrix.eye(4, ZZ), denom=8) b_bar_expected = B(to_col([0, 9, 5, 0]), denom=120) b_bar = C.reduce_element(b) assert b_bar == b_bar_expected a = A(to_col([1, 2, 3, 4])) raises(NotImplementedError, lambda: C.reduce_element(a)) C = B.submodule_from_matrix( DomainMatrix( [[5, 4, 3, 2], [0, 8, 7, 6], [0, 0, 11, 12], [0, 0, 0, 1]], (4, 4), ZZ).transpose()) raises(StructureError, lambda: C.reduce_element(b))
def test_round_two(): # Poly must be monic, irreducible, and over ZZ: raises(ValueError, lambda: round_two(Poly(3 * x ** 2 + 1))) raises(ValueError, lambda: round_two(Poly(x ** 2 - 1))) raises(ValueError, lambda: round_two(Poly(x ** 2 + QQ(1, 2)))) # Test on many fields: cases = ( # A couple of cyclotomic fields: (cyclotomic_poly(5), DomainMatrix.eye(4, QQ), 125), (cyclotomic_poly(7), DomainMatrix.eye(6, QQ), -16807), # A couple of quadratic fields (one 1 mod 4, one 3 mod 4): (x ** 2 - 5, DM([[1, (1, 2)], [0, (1, 2)]], QQ), 5), (x ** 2 - 7, DM([[1, 0], [0, 1]], QQ), 28), # Dedekind's example of a field with 2 as essential disc divisor: (x ** 3 + x ** 2 - 2 * x + 8, DM([[1, 0, 0], [0, 1, 0], [0, (1, 2), (1, 2)]], QQ).transpose(), -503), # A bunch of cubics with various forms for F -- all of these require # second or third enlargements. (Five of them require a third, while the rest require just a second.) # F = 2^2 (x**3 + 3 * x**2 - 4 * x + 4, DM([((1, 2), (1, 4), (1, 4)), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), -83), # F = 2^2 * 3 (x**3 + 3 * x**2 + 3 * x - 3, DM([((1, 2), 0, (1, 2)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -108), # F = 2^3 (x**3 + 5 * x**2 - x + 3, DM([((1, 4), 0, (3, 4)), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), -31), # F = 2^2 * 5 (x**3 + 5 * x**2 - 5 * x - 5, DM([((1, 2), 0, (1, 2)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), 1300), # F = 3^2 (x**3 + 3 * x**2 + 5, DM([((1, 3), (1, 3), (1, 3)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -135), # F = 3^3 (x**3 + 6 * x**2 + 3 * x - 1, DM([((1, 3), (1, 3), (1, 3)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), 81), # F = 2^2 * 3^2 (x**3 + 6 * x**2 + 4, DM([((1, 3), (2, 3), (1, 3)), (0, 1, 0), (0, 0, (1, 2))], QQ).transpose(), -108), # F = 2^3 * 7 (x**3 + 7 * x**2 + 7 * x - 7, DM([((1, 4), 0, (3, 4)), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), 49), # F = 2^2 * 13 (x**3 + 7 * x**2 - x + 5, DM([((1, 2), 0, (1, 2)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -2028), # F = 2^4 (x**3 + 7 * x**2 - 5 * x + 5, DM([((1, 4), 0, (3, 4)), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), -140), # F = 5^2 (x**3 + 4 * x**2 - 3 * x + 7, DM([((1, 5), (4, 5), (4, 5)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -175), # F = 7^2 (x**3 + 8 * x**2 + 5 * x - 1, DM([((1, 7), (6, 7), (2, 7)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), 49), # F = 2 * 5 * 7 (x**3 + 8 * x**2 - 2 * x + 6, DM([(1, 0, 0), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -14700), # F = 2^2 * 3 * 5 (x**3 + 6 * x**2 - 3 * x + 8, DM([(1, 0, 0), (0, (1, 4), (1, 4)), (0, 0, 1)], QQ).transpose(), -675), # F = 2 * 3^2 * 7 (x**3 + 9 * x**2 + 6 * x - 8, DM([(1, 0, 0), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), 3969), # F = 2^2 * 3^2 * 7 (x**3 + 15 * x**2 - 9 * x + 13, DM([((1, 6), (1, 3), (1, 6)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -5292), ) for f, B_exp, d_exp in cases: K = QQ.alg_field_from_poly(f) B = K.maximal_order().QQ_matrix d = K.discriminant() assert d == d_exp # The computed basis need not equal the expected one, but their quotient # must be unimodular: assert (B.inv()*B_exp).det()**2 == 1
def test_Submodule_is_compat_submodule(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) C = A.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ)) D = C.submodule_from_matrix(5 * DomainMatrix.eye(4, ZZ)) assert B.is_compat_submodule(C) is True assert B.is_compat_submodule(A) is False assert B.is_compat_submodule(D) is False
def test_Module_compat_col(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) col = to_col([1, 2, 3, 4]) row = col.transpose() assert A.is_compat_col(col) is True assert A.is_compat_col(row) is False assert A.is_compat_col(1) is False assert A.is_compat_col(DomainMatrix.eye(3, ZZ)[:, 0]) is False assert A.is_compat_col(DomainMatrix.eye(4, QQ)[:, 0]) is False assert A.is_compat_col(DomainMatrix.eye(4, ZZ)[:, 0]) is True
def test_Module_ancestors(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) C = B.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ)) D = B.submodule_from_matrix(5 * DomainMatrix.eye(4, ZZ)) assert C.ancestors(include_self=True) == [A, B, C] assert D.ancestors(include_self=True) == [A, B, D] assert C.power_basis_ancestor() == A assert C.nearest_common_ancestor(D) == B M = Module() assert M.power_basis_ancestor() is None
def test_Module_submodule_from_matrix(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) e = B(to_col([1, 2, 3, 4])) f = e.to_parent() assert f.col.flat() == [2, 4, 6, 8] # Matrix must be over ZZ: raises(ValueError, lambda: A.submodule_from_matrix(DomainMatrix.eye(4, QQ))) # Number of rows of matrix must equal number of generators of module A: raises(ValueError, lambda: A.submodule_from_matrix(2 * DomainMatrix.eye(5, ZZ)))
def test_check_formal_conditions_for_maximal_order(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) C = B.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ)) D = A.submodule_from_matrix(DomainMatrix.eye(4, ZZ)[:, :-1]) # Is a direct submodule of a power basis, but lacks 1 as first generator: raises(StructureError, lambda: _check_formal_conditions_for_maximal_order(B)) # Is not a direct submodule of a power basis: raises(StructureError, lambda: _check_formal_conditions_for_maximal_order(C)) # Is direct submod of pow basis, and starts with 1, but not sq/max rank/HNF: raises(StructureError, lambda: _check_formal_conditions_for_maximal_order(D))
def test_ModuleElement_to_ancestors(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) C = B.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ)) D = C.submodule_from_matrix(5 * DomainMatrix.eye(4, ZZ)) eD = D(0) eC = eD.to_parent() eB = eD.to_ancestor(B) eA = eD.over_power_basis() assert eC.module is C and eC.coeffs == [5, 0, 0, 0] assert eB.module is B and eB.coeffs == [15, 0, 0, 0] assert eA.module is A and eA.coeffs == [30, 0, 0, 0] a = A(0) raises(ValueError, lambda: a.to_parent())
def test_Submodule_repr(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ), denom=3) assert repr( B ) == 'Submodule[[2, 0, 0, 0], [0, 2, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2]]/3'
def test_supplement_a_subspace_1(): M = DM([[1, 7, 0], [2, 3, 4]], QQ).transpose() # First supplement over QQ: B = supplement_a_subspace(M) assert B[:, :2] == M assert B[:, 2] == DomainMatrix.eye(3, QQ).to_dense()[:, 0] # Now supplement over FF(7): M = M.convert_to(FF(7)) B = supplement_a_subspace(M) assert B[:, :2] == M # When we work mod 7, first col of M goes to [1, 0, 0], # so the supplementary vector cannot equal this, as it did # when we worked over QQ. Instead, we get the second std basis vector: assert B[:, 2] == DomainMatrix.eye(3, FF(7)).to_dense()[:, 1]
def test_Module_one(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) assert A.one().col.flat() == [1, 0, 0, 0] assert A.one().module == A assert B.one().col.flat() == [1, 0, 0, 0] assert B.one().module == A
def test_Module_call(): T = Poly(cyclotomic_poly(5, x)) B = PowerBasis(T) assert B(0).col.flat() == [1, 0, 0, 0] assert B(1).col.flat() == [0, 1, 0, 0] col = DomainMatrix.eye(4, ZZ)[:, 2] assert B(col).col == col raises(ValueError, lambda: B(-1))
def test_ModuleElement_compatibility(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) C = B.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ)) D = B.submodule_from_matrix(5 * DomainMatrix.eye(4, ZZ)) assert C(0).is_compat(C(1)) is True assert C(0).is_compat(D(0)) is False u, v = C(0).unify(D(0)) assert u.module is B and v.module is B assert C(C.represent(u)) == C(0) and D(D.represent(v)) == D(0) u, v = C(0).unify(C(1)) assert u == C(0) and v == C(1) U = Poly(cyclotomic_poly(7, x)) Z = PowerBasis(U) raises(UnificationFailed, lambda: C(0).unify(Z(1)))
def test_make_mod_elt(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) col = to_col([1, 2, 3, 4]) eA = make_mod_elt(A, col) eB = make_mod_elt(B, col) assert isinstance(eA, PowerBasisElement) assert not isinstance(eB, PowerBasisElement)
def test_Submodule_discard_before(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) B.compute_mult_tab() C = B.discard_before(2) assert C.parent == B.parent assert B.is_sq_maxrank_HNF() and not C.is_sq_maxrank_HNF() assert C.matrix == B.matrix[:, 2:] assert C.mult_tab() == {0: {0: [-2, -2], 1: [0, 0]}, 1: {1: [0, 0]}}
def test_ModuleElement_div(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) C = A.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ)) e = A(to_col([0, 2, 0, 0]), denom=3) f = A(to_col([0, 0, 0, 7]), denom=5) g = C(to_col([1, 1, 1, 1])) assert e // f == 10 * A(3) // 21 assert e // g == -2 * A(2) // 9 assert 3 // g == -A(1)
def test_Module_element_from_rational(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) rA = A.element_from_rational(QQ(22, 7)) rB = B.element_from_rational(QQ(22, 7)) assert rA.coeffs == [22, 0, 0, 0] assert rA.denom == 7 assert rA.module == A assert rB.coeffs == [22, 0, 0, 0] assert rB.denom == 7 assert rB.module == A
def test_ModuleElement_pow(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) C = A.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ)) e = A(to_col([0, 2, 0, 0]), denom=3) g = C(to_col([0, 0, 0, 1]), denom=2) assert e**3 == A(to_col([0, 0, 0, 8]), denom=27) assert g**2 == C(to_col([0, 3, 0, 0]), denom=4) assert e**0 == A(to_col([1, 0, 0, 0])) assert g**0 == A(to_col([1, 0, 0, 0])) assert e**1 == e assert g**1 == g
def test_Module_basis_elements(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) basis = B.basis_elements() bp = B.basis_element_pullbacks() for i, (e, p) in enumerate(zip(basis, bp)): c = [0] * 4 assert e.module == B assert p.module == A c[i] = 1 assert e == B(to_col(c)) c[i] = 2 assert p == A(to_col(c))
def test_find_min_poly(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) powers = [] m = find_min_poly(A(1), QQ, x=x, powers=powers) assert m == Poly(T, domain=QQ) assert len(powers) == 5 # powers list need not be passed m = find_min_poly(A(1), QQ, x=x) assert m == Poly(T, domain=QQ) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) raises(MissingUnityError, lambda: find_min_poly(B(1), QQ))
def test_ModuleElement_add(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) C = A.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ)) e = A(to_col([1, 2, 3, 4]), denom=6) f = A(to_col([5, 6, 7, 8]), denom=10) g = C(to_col([1, 1, 1, 1]), denom=2) assert e + f == A(to_col([10, 14, 18, 22]), denom=15) assert e - f == A(to_col([-5, -4, -3, -2]), denom=15) assert e + g == A(to_col([10, 11, 12, 13]), denom=6) assert e + QQ(7, 10) == A(to_col([26, 10, 15, 20]), denom=30) assert g + QQ(7, 10) == A(to_col([22, 15, 15, 15]), denom=10) U = Poly(cyclotomic_poly(7, x)) Z = PowerBasis(U) raises(TypeError, lambda: e + Z(0)) raises(TypeError, lambda: e + 3.14)
def test_ModuleElement_equiv(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) e = A(to_col([1, 2, 3, 4]), denom=1) f = A(to_col([3, 6, 9, 12]), denom=3) assert e.equiv(f) C = A.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ)) g = C(to_col([1, 2, 3, 4]), denom=1) h = A(to_col([3, 6, 9, 12]), denom=1) assert g.equiv(h) assert C(to_col([5, 0, 0, 0]), denom=7).equiv(QQ(15, 7)) U = Poly(cyclotomic_poly(7, x)) Z = PowerBasis(U) raises(UnificationFailed, lambda: e.equiv(Z(0))) assert e.equiv(3.14) is False
def test_ModuleElement_mul(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) C = A.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ)) e = A(to_col([0, 2, 0, 0]), denom=3) f = A(to_col([0, 0, 0, 7]), denom=5) g = C(to_col([0, 0, 0, 1]), denom=2) h = A(to_col([0, 0, 3, 1]), denom=7) assert e * f == A(to_col([-14, -14, -14, -14]), denom=15) assert e * g == A(to_col([-1, -1, -1, -1])) assert e * h == A(to_col([-2, -2, -2, 4]), denom=21) assert e * QQ(6, 5) == A(to_col([0, 4, 0, 0]), denom=5) assert (g * QQ(10, 21)).equiv(A(to_col([0, 0, 0, 5]), denom=7)) assert e // QQ(6, 5) == A(to_col([0, 5, 0, 0]), denom=9) U = Poly(cyclotomic_poly(7, x)) Z = PowerBasis(U) raises(TypeError, lambda: e * Z(0)) raises(TypeError, lambda: e * 3.14) raises(TypeError, lambda: e // 3.14) raises(ZeroDivisionError, lambda: e // 0)
def eye(cls, n, gens): return cls.from_dm(DomainMatrix.eye(n, QQ[gens]))
def test_Submodule_eq(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) C = A.submodule_from_matrix(6 * DomainMatrix.eye(4, ZZ), denom=3) assert C == B
def _eval_eye(cls, rows, cols): rep = DomainMatrix.eye((rows, cols), ZZ) return cls._fromrep(rep)
def test_Module_starts_with_unity(): T = Poly(cyclotomic_poly(5, x)) A = PowerBasis(T) B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ)) assert A.starts_with_unity() is True assert B.starts_with_unity() is False