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