def test_PowerBasis_mult_tab():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
M = A.mult_tab()
exp = {
0: {
0: [1, 0, 0, 0],
1: [0, 1, 0, 0],
2: [0, 0, 1, 0],
3: [0, 0, 0, 1]
},
1: {
1: [0, 0, 1, 0],
2: [0, 0, 0, 1],
3: [-1, -1, -1, -1]
},
2: {
2: [-1, -1, -1, -1],
3: [1, 0, 0, 0]
},
3: {
3: [0, 1, 0, 0]
}
}
# We get the table we expect:
assert M == exp
# And all entries are of expected type:
assert all(is_int(c) for u in M for v in M[u] for c in M[u][v])
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_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_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_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_PowerBasis_represent():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
col = to_col([1, 2, 3, 4])
a = A(col)
assert A.represent(a) == col
b = A(col, denom=2)
raises(ClosureFailure, lambda: A.represent(b))
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_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_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_PowerBasis_element_from_poly():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
f = Poly(1 + 2 * x)
g = Poly(x**4)
h = Poly(0, x)
assert A.element_from_poly(f).coeffs == [1, 2, 0, 0]
assert A.element_from_poly(g).coeffs == [-1, -1, -1, -1]
assert A.element_from_poly(h).coeffs == [0, 0, 0, 0]
def test_Module_whole_submodule():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.whole_submodule()
e = B(to_col([1, 2, 3, 4]))
f = e.to_parent()
assert f.col.flat() == [1, 2, 3, 4]
e0, e1, e2, e3 = B(0), B(1), B(2), B(3)
assert e2 * e3 == e0
assert e3**2 == e1
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_ModuleElement_eq():
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 == f
U = Poly(cyclotomic_poly(7, x))
Z = PowerBasis(U)
assert e != Z(0)
assert e != 3.14
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_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_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_ModuleElement_mod():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(to_col([1, 15, 8, 0]), denom=2)
assert e % 7 == A(to_col([1, 1, 8, 0]), denom=2)
assert e % QQ(1, 2) == A.zero()
assert e % QQ(1, 3) == A(to_col([1, 1, 0, 0]), denom=6)
B = A.submodule_from_gens([A(0), 5 * A(1), 3 * A(2), A(3)])
assert e % B == A(to_col([1, 5, 2, 0]), denom=2)
C = B.whole_submodule()
raises(TypeError, lambda: e % C)
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_Module_submodule_from_gens():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
gens = [2 * A(0), 2 * A(1), 6 * A(0), 6 * A(1)]
B = A.submodule_from_gens(gens)
# Because the 3rd and 4th generators do not add anything new, we expect
# the cols of the matrix of B to just reproduce the first two gens:
M = gens[0].column().hstack(gens[1].column())
assert B.matrix == M
# At least one generator must be provided:
raises(ValueError, lambda: A.submodule_from_gens([]))
# All generators must belong to A:
raises(ValueError, lambda: A.submodule_from_gens([3 * A(0), B(0)]))
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_Submodule_add():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(DomainMatrix([
[4, 0, 0, 0],
[0, 4, 0, 0],
], (2, 4), ZZ).transpose(),
denom=6)
C = A.submodule_from_matrix(DomainMatrix([
[0, 10, 0, 0],
[0, 0, 7, 0],
], (2, 4), ZZ).transpose(),
denom=15)
D = A.submodule_from_matrix(DomainMatrix([
[20, 0, 0, 0],
[0, 20, 0, 0],
[0, 0, 14, 0],
], (3, 4), ZZ).transpose(),
denom=30)
assert B + C == D
U = Poly(cyclotomic_poly(7, x))
Z = PowerBasis(U)
Y = Z.submodule_from_gens([Z(0), Z(1)])
raises(TypeError, lambda: B + Y)
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_EndomorphismRing_represent():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
R = A.endomorphism_ring()
phi = R.inner_endomorphism(A(1))
col = R.represent(phi)
assert col.transpose() == DomainMatrix(
[[0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, -1, -1, -1, -1]], (1, 16), ZZ)
B = A.submodule_from_matrix(DomainMatrix.zeros((4, 0), ZZ))
S = B.endomorphism_ring()
psi = S.inner_endomorphism(A(1))
col = S.represent(psi)
assert col == DomainMatrix([], (0, 0), ZZ)
raises(NotImplementedError, lambda: R.represent(3.14))
def test_PowerBasisElement_polys():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(to_col([1, 15, 8, 0]), denom=2)
assert e.numerator(x=zeta) == Poly(8 * zeta**2 + 15 * zeta + 1, domain=ZZ)
assert e.poly(x=zeta) == Poly(4 * zeta**2 + QQ(15, 2) * zeta + QQ(1, 2),
domain=QQ)
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_ModuleHomomorphism_matrix():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
phi = ModuleEndomorphism(A, lambda a: a**2)
M = phi.matrix()
assert M == DomainMatrix(
[[1, 0, -1, 0], [0, 0, -1, 1], [0, 1, -1, 0], [0, 0, -1, 0]], (4, 4),
ZZ)
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_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)
