def test_exceptions():
I = QQ.old_poly_ring(x).ideal(x)
J = QQ.old_poly_ring(y).ideal(1)
raises(ValueError, lambda: I.union(x))
raises(ValueError, lambda: I + J)
raises(ValueError, lambda: I * J)
raises(ValueError, lambda: I.union(J))
assert (I == J) is False
assert I != J
def test_intersection():
R = QQ.old_poly_ring(x, y, z)
# SCA, example 1.8.11
assert R.ideal(x, y).intersect(R.ideal(y ** 2, z)) == R.ideal(y ** 2, y * z, x * z)
assert R.ideal(x, y).intersect(R.ideal()).is_zero()
R = QQ.old_poly_ring(x, y, z, order="ilex")
assert R.ideal(x, y).intersect(R.ideal(y ** 2 + y ** 2 * z, z + z * x ** 3 * y)) == R.ideal(y ** 2, y * z, x * z)
def test_quotient():
# SCA, example 2.8.6
R = QQ.old_poly_ring(x, y, z)
F = R.free_module(2)
assert F.submodule([x*y, x*z], [y*z, x*y]).module_quotient(
F.submodule([y, z], [z, y])) == QQ.old_poly_ring(x, y, z).ideal(x**2*y**2 - x*y*z**2)
assert F.submodule([x, y]).module_quotient(F.submodule()).is_whole_ring()
M = F.submodule([x**2, x**2], [y**2, y**2])
N = F.submodule([x + y, x + y])
q, rel = M.module_quotient(N, relations=True)
assert q == R.ideal(y**2, x - y)
for i, g in enumerate(q.gens):
assert g*N.gens[0] == sum(c*x for c, x in zip(rel[i], M.gens))
def test_FreeModule():
M1 = FreeModule(QQ.old_poly_ring(x), 2)
assert M1 == FreeModule(QQ.old_poly_ring(x), 2)
assert M1 != FreeModule(QQ.old_poly_ring(y), 2)
assert M1 != FreeModule(QQ.old_poly_ring(x), 3)
M2 = FreeModule(QQ.old_poly_ring(x, order="ilex"), 2)
assert [x, 1] in M1
assert [x] not in M1
assert [2, y] not in M1
assert [1/(x + 1), 2] not in M1
e = M1.convert([x, x**2 + 1])
X = QQ.old_poly_ring(x).convert(x)
assert e == [X, X**2 + 1]
assert e == [x, x**2 + 1]
assert 2*e == [2*x, 2*x**2 + 2]
assert e*2 == [2*x, 2*x**2 + 2]
assert e/2 == [x/2, (x**2 + 1)/2]
assert x*e == [x**2, x**3 + x]
assert e*x == [x**2, x**3 + x]
assert X*e == [x**2, x**3 + x]
assert e*X == [x**2, x**3 + x]
assert [x, 1] in M2
assert [x] not in M2
assert [2, y] not in M2
assert [1/(x + 1), 2] in M2
e = M2.convert([x, x**2 + 1])
X = QQ.old_poly_ring(x, order="ilex").convert(x)
assert e == [X, X**2 + 1]
assert e == [x, x**2 + 1]
assert 2*e == [2*x, 2*x**2 + 2]
assert e*2 == [2*x, 2*x**2 + 2]
assert e/2 == [x/2, (x**2 + 1)/2]
assert x*e == [x**2, x**3 + x]
assert e*x == [x**2, x**3 + x]
assert e/(1 + x) == [x/(1 + x), (x**2 + 1)/(1 + x)]
assert X*e == [x**2, x**3 + x]
assert e*X == [x**2, x**3 + x]
M3 = FreeModule(QQ.old_poly_ring(x, y), 2)
assert M3.convert(e) == M3.convert([x, x**2 + 1])
assert not M3.is_submodule(0)
assert not M3.is_zero()
raises(NotImplementedError, lambda: ZZ.old_poly_ring(x).free_module(2))
raises(NotImplementedError, lambda: FreeModulePolyRing(ZZ, 2))
raises(CoercionFailed, lambda: M1.convert(QQ.old_poly_ring(x).free_module(3)
.convert([1, 2, 3])))
raises(CoercionFailed, lambda: M3.convert(1))
def test_in_terms_of_generators():
R = QQ.old_poly_ring(x, order="ilex")
M = R.free_module(2).submodule([2*x, 0], [1, 2])
assert M.in_terms_of_generators(
[x, x]) == [R.convert(S(1)/4), R.convert(x/2)]
raises(ValueError, lambda: M.in_terms_of_generators([1, 0]))
M = R.free_module(2) / ([x, 0], [1, 1])
SM = M.submodule([1, x])
assert SM.in_terms_of_generators([2, 0]) == [R.convert(-2/(x - 1))]
R = QQ.old_poly_ring(x, y) / [x**2 - y**2]
M = R.free_module(2)
SM = M.submodule([x, 0], [0, y])
assert SM.in_terms_of_generators(
[x**2, x**2]) == [R.convert(x), R.convert(y)]
def test_QuotientModule():
R = QQ.old_poly_ring(x)
F = R.free_module(3)
N = F.submodule([1, x, x**2])
M = F / N
assert M != F
assert M != N
assert M == F / [(1, x, x**2)]
assert not M.is_zero()
assert (F / F.basis()).is_zero()
SQ = F.submodule([1, x, x**2], [2, 0, 0]) / N
assert SQ == M.submodule([2, x, x**2])
assert SQ != M.submodule([2, 1, 0])
assert SQ != M
assert M.is_submodule(SQ)
assert not SQ.is_full_module()
raises(ValueError, lambda: N / F)
raises(ValueError, lambda: F.submodule([2, 0, 0]) / N)
raises(ValueError, lambda: R.free_module(2) / F)
raises(CoercionFailed, lambda: F.convert(M.convert([1, x, x**2])))
M1 = F / [[1, 1, 1]]
M2 = M1.submodule([1, 0, 0], [0, 1, 0])
assert M1 == M2
def test_SubModulePolyRing_nontriv_global():
R = QQ.old_poly_ring(x, y, z)
F = R.free_module(1)
def contains(I, f):
return F.submodule(*[[g] for g in I]).contains([f])
assert contains([x, y], x)
assert contains([x, y], x + y)
assert not contains([x, y], 1)
assert not contains([x, y], z)
assert contains([x**2 + y, x**2 + x], x - y)
assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2)
assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**3)
assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4)
assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y**2)
assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4 + y**3 + 2*z*y*x)
assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y*z)
assert contains([x, 1 + x + y, 5 - 7*y], 1)
assert contains(
[x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z],
x**3)
assert not contains(
[x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z],
x**2 + y**2)
# compare local order
assert not contains([x*(1 + x + y), y*(1 + z)], x)
assert not contains([x*(1 + x + y), y*(1 + z)], x + y)
def test_SubModulePolyRing_local():
R = QQ.old_poly_ring(x, y, order=ilex)
F = R.free_module(3)
Fd = F.submodule([1 + x, 0, 0], [1 + y, 2 + 2*y, 0], [1, 2, 3])
M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])
assert F == Fd
assert Fd == F
assert F != M
assert M != F
assert Fd != M
assert M != Fd
assert Fd == F.submodule(*F.basis())
assert Fd.is_full_module()
assert not M.is_full_module()
assert not Fd.is_zero()
assert not M.is_zero()
assert Fd.submodule().is_zero()
assert M.contains([x**2 + y**2 + x, 1 + y, 1])
assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
assert M.contains([y**2, 1 - x*y, -x])
assert F.submodule([1 + x, 0, 0]) == F.submodule([1, 0, 0])
assert F.submodule(
[1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1 + x*y])) == F
raises(ValueError, lambda: M.submodule([1, 0, 0]))
def test_SubModulePolyRing_local():
R = QQ.old_poly_ring(x, y, order=ilex)
F = R.free_module(3)
Fd = F.submodule([1 + x, 0, 0], [1 + y, 2 + 2 * y, 0], [1, 2, 3])
M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])
assert F == Fd
assert Fd == F
assert F != M
assert M != F
assert Fd != M
assert M != Fd
assert Fd == F.submodule(*F.basis())
assert Fd.is_full_module()
assert not M.is_full_module()
assert not Fd.is_zero()
assert not M.is_zero()
assert Fd.submodule().is_zero()
assert M.contains([x**2 + y**2 + x, 1 + y, 1])
assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
assert M.contains([y**2, 1 - x * y, -x])
assert F.submodule([1 + x, 0, 0]) == F.submodule([1, 0, 0])
assert F.submodule([1, 0, 0],
[0, 1, 0]).union(F.submodule([0, 0, 1 + x * y])) == F
raises(ValueError, lambda: M.submodule([1, 0, 0]))
def test_SubModulePolyRing_nontriv_global():
R = QQ.old_poly_ring(x, y, z)
F = R.free_module(1)
def contains(I, f):
return F.submodule(*[[g] for g in I]).contains([f])
assert contains([x, y], x)
assert contains([x, y], x + y)
assert not contains([x, y], 1)
assert not contains([x, y], z)
assert contains([x**2 + y, x**2 + x], x - y)
assert not contains([x + y + z, x * y + x * z + y * z, x * y * z], x**2)
assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x**3)
assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x**4)
assert not contains([x + y + z, x * y + x * z + y * z, x * y * z],
x * y**2)
assert contains([x + y + z, x * y + x * z + y * z, x * y * z],
x**4 + y**3 + 2 * z * y * x)
assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x * y * z)
assert contains([x, 1 + x + y, 5 - 7 * y], 1)
assert contains([
x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2 * y + x**2 * z + y**2 * z
], x**3)
assert not contains([
x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2 * y + x**2 * z + y**2 * z
], x**2 + y**2)
# compare local order
assert not contains([x * (1 + x + y), y * (1 + z)], x)
assert not contains([x * (1 + x + y), y * (1 + z)], x + y)
def test_nontriv_global():
R = QQ.old_poly_ring(x, y, z)
def contains(I, f):
return R.ideal(*I).contains(f)
assert contains([x, y], x)
assert contains([x, y], x + y)
assert not contains([x, y], 1)
assert not contains([x, y], z)
assert contains([x ** 2 + y, x ** 2 + x], x - y)
assert not contains([x + y + z, x * y + x * z + y * z, x * y * z], x ** 2)
assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x ** 3)
assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x ** 4)
assert not contains([x + y + z, x * y + x * z + y * z, x * y * z], x * y ** 2)
assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x ** 4 + y ** 3 + 2 * z * y * x)
assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x * y * z)
assert contains([x, 1 + x + y, 5 - 7 * y], 1)
assert contains([x ** 3 + y ** 3, y ** 3 + z ** 3, z ** 3 + x ** 3, x ** 2 * y + x ** 2 * z + y ** 2 * z], x ** 3)
assert not contains(
[x ** 3 + y ** 3, y ** 3 + z ** 3, z ** 3 + x ** 3, x ** 2 * y + x ** 2 * z + y ** 2 * z], x ** 2 + y ** 2
)
# compare local order
assert not contains([x * (1 + x + y), y * (1 + z)], x)
assert not contains([x * (1 + x + y), y * (1 + z)], x + y)
def test_QuotientModule():
R = QQ.old_poly_ring(x)
F = R.free_module(3)
N = F.submodule([1, x, x**2])
M = F/N
assert M != F
assert M != N
assert M == F / [(1, x, x**2)]
assert not M.is_zero()
assert (F / F.basis()).is_zero()
SQ = F.submodule([1, x, x**2], [2, 0, 0]) / N
assert SQ == M.submodule([2, x, x**2])
assert SQ != M.submodule([2, 1, 0])
assert SQ != M
assert M.is_submodule(SQ)
assert not SQ.is_full_module()
raises(ValueError, lambda: N/F)
raises(ValueError, lambda: F.submodule([2, 0, 0]) / N)
raises(ValueError, lambda: R.free_module(2)/F)
raises(CoercionFailed, lambda: F.convert(M.convert([1, x, x**2])))
M1 = F / [[1, 1, 1]]
M2 = M1.submodule([1, 0, 0], [0, 1, 0])
assert M1 == M2
def test_QuotientModuleElement():
R = QQ.old_poly_ring(x)
F = R.free_module(3)
N = F.submodule([1, x, x**2])
M = F/N
e = M.convert([x**2, 2, 0])
assert M.convert([x + 1, x**2 + x, x**3 + x**2]) == 0
assert e == [x**2, 2, 0] + N == F.convert([x**2, 2, 0]) + N == \
M.convert(F.convert([x**2, 2, 0]))
assert M.convert([x**2 + 1, 2*x + 2, x**2]) == e + [0, x, 0] == \
e + M.convert([0, x, 0]) == e + F.convert([0, x, 0])
assert M.convert([x**2 + 1, 2, x**2]) == e - [0, x, 0] == \
e - M.convert([0, x, 0]) == e - F.convert([0, x, 0])
assert M.convert([0, 2, 0]) == M.convert([x**2, 4, 0]) - e == \
[x**2, 4, 0] - e == F.convert([x**2, 4, 0]) - e
assert M.convert([x**3 + x**2, 2*x + 2, 0]) == (1 + x)*e == \
R.convert(1 + x)*e == e*(1 + x) == e*R.convert(1 + x)
assert -e == [-x**2, -2, 0]
f = [x, x, 0] + N
assert M.convert([1, 1, 0]) == f / x == f / R.convert(x)
M2 = F/[(2, 2*x, 2*x**2), (0, 0, 1)]
G = R.free_module(2)
M3 = G/[[1, x]]
M4 = F.submodule([1, x, x**2], [1, 0, 0]) / N
raises(CoercionFailed, lambda: M.convert(G.convert([1, x])))
raises(CoercionFailed, lambda: M.convert(M3.convert([1, x])))
raises(CoercionFailed, lambda: M.convert(M2.convert([1, x, x])))
assert M2.convert(M.convert([2, x, x**2])) == [2, x, 0]
assert M.convert(M4.convert([2, 0, 0])) == [2, 0, 0]
def test_reduction():
from sympy.polys.distributedmodules import sdm_nf_buchberger_reduced
R = QQ.old_poly_ring(x, y)
I = R.ideal(x**5, y)
e = R.convert(x**3 + y**2)
assert I.reduce_element(e) == e
assert I.reduce_element(e, NF=sdm_nf_buchberger_reduced) == R.convert(x**3)
def test_nontriv_global():
R = QQ.old_poly_ring(x, y, z)
def contains(I, f):
return R.ideal(*I).contains(f)
assert contains([x, y], x)
assert contains([x, y], x + y)
assert not contains([x, y], 1)
assert not contains([x, y], z)
assert contains([x**2 + y, x**2 + x], x - y)
assert not contains([x + y + z, x * y + x * z + y * z, x * y * z], x**2)
assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x**3)
assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x**4)
assert not contains([x + y + z, x * y + x * z + y * z, x * y * z],
x * y**2)
assert contains([x + y + z, x * y + x * z + y * z, x * y * z],
x**4 + y**3 + 2 * z * y * x)
assert contains([x + y + z, x * y + x * z + y * z, x * y * z], x * y * z)
assert contains([x, 1 + x + y, 5 - 7 * y], 1)
assert contains([
x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2 * y + x**2 * z + y**2 * z
], x**3)
assert not contains([
x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2 * y + x**2 * z + y**2 * z
], x**2 + y**2)
# compare local order
assert not contains([x * (1 + x + y), y * (1 + z)], x)
assert not contains([x * (1 + x + y), y * (1 + z)], x + y)
def test_QuotientModuleElement():
R = QQ.old_poly_ring(x)
F = R.free_module(3)
N = F.submodule([1, x, x**2])
M = F / N
e = M.convert([x**2, 2, 0])
assert M.convert([x + 1, x**2 + x, x**3 + x**2]) == 0
assert (e == [x**2, 2, 0] + N == F.convert([x**2, 2, 0]) + N == M.convert(
F.convert([x**2, 2, 0])))
assert (M.convert([x**2 + 1, 2 * x + 2, x**2]) == e + [0, x, 0] ==
e + M.convert([0, x, 0]) == e + F.convert([0, x, 0]))
assert (M.convert([x**2 + 1, 2, x**2]) == e - [0, x, 0] ==
e - M.convert([0, x, 0]) == e - F.convert([0, x, 0]))
assert (M.convert([0, 2, 0]) == M.convert([x**2, 4, 0]) - e ==
[x**2, 4, 0] - e == F.convert([x**2, 4, 0]) - e)
assert (M.convert([x**3 + x**2, 2 * x + 2, 0]) ==
(1 + x) * e == R.convert(1 + x) * e == e *
(1 + x) == e * R.convert(1 + x))
assert -e == [-(x**2), -2, 0]
f = [x, x, 0] + N
assert M.convert([1, 1, 0]) == f / x == f / R.convert(x)
M2 = F / [(2, 2 * x, 2 * x**2), (0, 0, 1)]
G = R.free_module(2)
M3 = G / [[1, x]]
M4 = F.submodule([1, x, x**2], [1, 0, 0]) / N
raises(CoercionFailed, lambda: M.convert(G.convert([1, x])))
raises(CoercionFailed, lambda: M.convert(M3.convert([1, x])))
raises(CoercionFailed, lambda: M.convert(M2.convert([1, x, x])))
assert M2.convert(M.convert([2, x, x**2])) == [2, x, 0]
assert M.convert(M4.convert([2, 0, 0])) == [2, 0, 0]
def test_reduction():
from sympy.polys.distributedmodules import sdm_nf_buchberger_reduced
R = QQ.old_poly_ring(x, y)
I = R.ideal(x**5, y)
e = R.convert(x**3 + y**2)
assert I.reduce_element(e) == e
assert I.reduce_element(e, NF=sdm_nf_buchberger_reduced) == R.convert(x**3)
def test_in_terms_of_generators():
R = QQ.old_poly_ring(x, order="ilex")
M = R.free_module(2).submodule([2 * x, 0], [1, 2])
assert M.in_terms_of_generators(
[x, x]) == [R.convert(S(1) / 4), R.convert(x / 2)]
raises(ValueError, lambda: M.in_terms_of_generators([1, 0]))
M = R.free_module(2) / ([x, 0], [1, 1])
SM = M.submodule([1, x])
assert SM.in_terms_of_generators([2, 0]) == [R.convert(-2 / (x - 1))]
R = QQ.old_poly_ring(x, y) / [x**2 - y**2]
M = R.free_module(2)
SM = M.submodule([x, 0], [0, y])
assert SM.in_terms_of_generators([x**2,
x**2]) == [R.convert(x),
R.convert(y)]
def test_syzygy():
R = QQ.old_poly_ring(x, y, z)
M = R.free_module(1).submodule([x*y], [y*z], [x*z])
S = R.free_module(3).submodule([0, x, -y], [z, -x, 0])
assert M.syzygy_module() == S
M2 = M / ([x*y*z],)
S2 = R.free_module(3).submodule([z, 0, 0], [0, x, 0], [0, 0, y])
assert M2.syzygy_module() == S2
F = R.free_module(3)
assert F.submodule(*F.basis()).syzygy_module() == F.submodule()
R2 = QQ.old_poly_ring(x, y, z) / [x*y*z]
M3 = R2.free_module(1).submodule([x*y], [y*z], [x*z])
S3 = R2.free_module(3).submodule([z, 0, 0], [0, x, 0], [0, 0, y])
assert M3.syzygy_module() == S3
def test_syzygy():
R = QQ.old_poly_ring(x, y, z)
M = R.free_module(1).submodule([x * y], [y * z], [x * z])
S = R.free_module(3).submodule([0, x, -y], [z, -x, 0])
assert M.syzygy_module() == S
M2 = M / ([x * y * z], )
S2 = R.free_module(3).submodule([z, 0, 0], [0, x, 0], [0, 0, y])
assert M2.syzygy_module() == S2
F = R.free_module(3)
assert F.submodule(*F.basis()).syzygy_module() == F.submodule()
R2 = QQ.old_poly_ring(x, y, z) / [x * y * z]
M3 = R2.free_module(1).submodule([x * y], [y * z], [x * z])
S3 = R2.free_module(3).submodule([z, 0, 0], [0, x, 0], [0, 0, y])
assert M3.syzygy_module() == S3
def test_module_mul():
R = QQ.old_poly_ring(x)
M = R.free_module(2)
S1 = M.submodule([x, 0], [0, x])
S2 = M.submodule([x**2, 0], [0, x**2])
I = R.ideal(x)
assert I * M == M * I == S1 == x * M == M * x
assert I * S1 == S2 == x * S1
def test_module_mul():
R = QQ.old_poly_ring(x)
M = R.free_module(2)
S1 = M.submodule([x, 0], [0, x])
S2 = M.submodule([x**2, 0], [0, x**2])
I = R.ideal(x)
assert I*M == M*I == S1 == x*M == M*x
assert I*S1 == S2 == x*S1
def test_ModulesQuotientRing():
R = QQ.old_poly_ring(x, y, order=(("lex", x), ("ilex", y))) / [x**2 + 1]
M1 = R.free_module(2)
assert M1 == R.free_module(2)
assert M1 != QQ.old_poly_ring(x).free_module(2)
assert M1 != R.free_module(3)
assert [x, 1] in M1
assert [x] not in M1
assert [1/(R.convert(x) + 1), 2] in M1
assert [1, 2/(1 + y)] in M1
assert [1, 2/y] not in M1
assert M1.convert([x**2, y]) == [-1, y]
F = R.free_module(3)
Fd = F.submodule([x**2, 0, 0], [1, 2, 0], [1, 2, 3])
M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])
assert F == Fd
assert Fd == F
assert F != M
assert M != F
assert Fd != M
assert M != Fd
assert Fd == F.submodule(*F.basis())
assert Fd.is_full_module()
assert not M.is_full_module()
assert not Fd.is_zero()
assert not M.is_zero()
assert Fd.submodule().is_zero()
assert M.contains([x**2 + y**2 + x, -x**2 + y, 1])
assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
assert M.contains([y**2, 1 - x*y, -x])
assert F.submodule([x, 0, 0]) == F.submodule([1, 0, 0])
assert not F.submodule([y, 0, 0]) == F.submodule([1, 0, 0])
assert F.submodule([1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1])) == F
assert not M.is_submodule(0)
def test_ModulesQuotientRing():
R = QQ.old_poly_ring(x, y, order=(("lex", x), ("ilex", y))) / [x**2 + 1]
M1 = R.free_module(2)
assert M1 == R.free_module(2)
assert M1 != QQ.old_poly_ring(x).free_module(2)
assert M1 != R.free_module(3)
assert [x, 1] in M1
assert [x] not in M1
assert [1 / (R.convert(x) + 1), 2] in M1
assert [1, 2 / (1 + y)] in M1
assert [1, 2 / y] not in M1
assert M1.convert([x**2, y]) == [-1, y]
F = R.free_module(3)
Fd = F.submodule([x**2, 0, 0], [1, 2, 0], [1, 2, 3])
M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])
assert F == Fd
assert Fd == F
assert F != M
assert M != F
assert Fd != M
assert M != Fd
assert Fd == F.submodule(*F.basis())
assert Fd.is_full_module()
assert not M.is_full_module()
assert not Fd.is_zero()
assert not M.is_zero()
assert Fd.submodule().is_zero()
assert M.contains([x**2 + y**2 + x, -x**2 + y, 1])
assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
assert M.contains([y**2, 1 - x * y, -x])
assert F.submodule([x, 0, 0]) == F.submodule([1, 0, 0])
assert not F.submodule([y, 0, 0]) == F.submodule([1, 0, 0])
assert F.submodule([1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1])) == F
assert not M.is_submodule(0)
def test_intersection():
# SCA, example 2.8.5
F = QQ.old_poly_ring(x, y).free_module(2)
M1 = F.submodule([x, y], [y, 1])
M2 = F.submodule([0, y - 1], [x, 1], [y, x])
I = F.submodule([x, y], [y**2 - y, y - 1], [x * y + y, x + 1])
I1, rel1, rel2 = M1.intersect(M2, relations=True)
assert I1 == M2.intersect(M1) == I
for i, g in enumerate(I1.gens):
assert g == sum(c*x for c, x in zip(rel1[i], M1.gens)) \
== sum(d*y for d, y in zip(rel2[i], M2.gens))
assert F.submodule([x, y]).intersect(F.submodule([y, x])).is_zero()
def test_intersection():
# SCA, example 2.8.5
F = QQ.old_poly_ring(x, y).free_module(2)
M1 = F.submodule([x, y], [y, 1])
M2 = F.submodule([0, y - 1], [x, 1], [y, x])
I = F.submodule([x, y], [y**2 - y, y - 1], [x*y + y, x + 1])
I1, rel1, rel2 = M1.intersect(M2, relations=True)
assert I1 == M2.intersect(M1) == I
for i, g in enumerate(I1.gens):
assert g == sum(c*x for c, x in zip(rel1[i], M1.gens)) \
== sum(d*y for d, y in zip(rel2[i], M2.gens))
assert F.submodule([x, y]).intersect(F.submodule([y, x])).is_zero()
def test_nontriv_local():
R = QQ.old_poly_ring(x, y, z, order=ilex)
def contains(I, f):
return R.ideal(*I).contains(f)
assert contains([x, y], x)
assert contains([x, y], x + y)
assert not contains([x, y], 1)
assert not contains([x, y], z)
assert contains([x ** 2 + y, x ** 2 + x], x - y)
assert not contains([x + y + z, x * y + x * z + y * z, x * y * z], x ** 2)
assert contains([x * (1 + x + y), y * (1 + z)], x)
assert contains([x * (1 + x + y), y * (1 + z)], x + y)
def test_nontriv_local():
R = QQ.old_poly_ring(x, y, z, order=ilex)
def contains(I, f):
return R.ideal(*I).contains(f)
assert contains([x, y], x)
assert contains([x, y], x + y)
assert not contains([x, y], 1)
assert not contains([x, y], z)
assert contains([x**2 + y, x**2 + x], x - y)
assert not contains([x + y + z, x * y + x * z + y * z, x * y * z], x**2)
assert contains([x * (1 + x + y), y * (1 + z)], x)
assert contains([x * (1 + x + y), y * (1 + z)], x + y)
def test_SubModulePolyRing_nontriv_local():
R = QQ.old_poly_ring(x, y, z, order=ilex)
F = R.free_module(1)
def contains(I, f):
return F.submodule(*[[g] for g in I]).contains([f])
assert contains([x, y], x)
assert contains([x, y], x + y)
assert not contains([x, y], 1)
assert not contains([x, y], z)
assert contains([x**2 + y, x**2 + x], x - y)
assert not contains([x + y + z, x * y + x * z + y * z, x * y * z], x**2)
assert contains([x * (1 + x + y), y * (1 + z)], x)
assert contains([x * (1 + x + y), y * (1 + z)], x + y)
def test_SubModulePolyRing_nontriv_local():
R = QQ.old_poly_ring(x, y, z, order=ilex)
F = R.free_module(1)
def contains(I, f):
return F.submodule(*[[g] for g in I]).contains([f])
assert contains([x, y], x)
assert contains([x, y], x + y)
assert not contains([x, y], 1)
assert not contains([x, y], z)
assert contains([x**2 + y, x**2 + x], x - y)
assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2)
assert contains([x*(1 + x + y), y*(1 + z)], x)
assert contains([x*(1 + x + y), y*(1 + z)], x + y)
def test_ideal_operations():
R = QQ.old_poly_ring(x, y)
I = R.ideal(x)
J = R.ideal(y)
S = R.ideal(x * y)
T = R.ideal(x, y)
assert not (I == J)
assert I == I
assert I.union(J) == T
assert I + J == T
assert I + T == T
assert not I.subset(T)
assert T.subset(I)
assert I.product(J) == S
assert I * J == S
assert x * J == S
assert I * y == S
assert R.convert(x) * J == S
assert I * R.convert(y) == S
assert not I.is_zero()
assert not J.is_whole_ring()
assert R.ideal(x ** 2 + 1, x).is_whole_ring()
assert R.ideal() == R.ideal(0)
assert R.ideal().is_zero()
assert T.contains(x * y)
assert T.subset([x, y])
assert T.in_terms_of_generators(x) == [R(1), R(0)]
assert T ** 0 == R.ideal(1)
assert T ** 1 == T
assert T ** 2 == R.ideal(x ** 2, y ** 2, x * y)
assert I ** 5 == R.ideal(x ** 5)
def test_ideal_operations():
R = QQ.old_poly_ring(x, y)
I = R.ideal(x)
J = R.ideal(y)
S = R.ideal(x * y)
T = R.ideal(x, y)
assert not (I == J)
assert I == I
assert I.union(J) == T
assert I + J == T
assert I + T == T
assert not I.subset(T)
assert T.subset(I)
assert I.product(J) == S
assert I * J == S
assert x * J == S
assert I * y == S
assert R.convert(x) * J == S
assert I * R.convert(y) == S
assert not I.is_zero()
assert not J.is_whole_ring()
assert R.ideal(x**2 + 1, x).is_whole_ring()
assert R.ideal() == R.ideal(0)
assert R.ideal().is_zero()
assert T.contains(x * y)
assert T.subset([x, y])
assert T.in_terms_of_generators(x) == [R(1), R(0)]
assert T**0 == R.ideal(1)
assert T**1 == T
assert T**2 == R.ideal(x**2, y**2, x * y)
assert I**5 == R.ideal(x**5)
def test_SubModulePolyRing_global():
R = QQ.old_poly_ring(x, y)
F = R.free_module(3)
Fd = F.submodule([1, 0, 0], [1, 2, 0], [1, 2, 3])
M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])
assert F == Fd
assert Fd == F
assert F != M
assert M != F
assert Fd != M
assert M != Fd
assert Fd == F.submodule(*F.basis())
assert Fd.is_full_module()
assert not M.is_full_module()
assert not Fd.is_zero()
assert not M.is_zero()
assert Fd.submodule().is_zero()
assert M.contains([x**2 + y**2 + x, 1 + y, 1])
assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
assert M.contains([y**2, 1 - x * y, -x])
assert not F.submodule([1 + x, 0, 0]) == F.submodule([1, 0, 0])
assert F.submodule([1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1])) == F
assert not M.is_submodule(0)
m = F.convert([x**2 + y**2, 1, 0])
n = M.convert(m)
assert m.module is F
assert n.module is M
raises(ValueError, lambda: M.submodule([1, 0, 0]))
raises(TypeError, lambda: M.union(1))
raises(ValueError, lambda: M.union(R.free_module(1).submodule([x])))
assert F.submodule([x, x, x]) != F.submodule([x, x, x], order="ilex")
def test_SubModulePolyRing_global():
R = QQ.old_poly_ring(x, y)
F = R.free_module(3)
Fd = F.submodule([1, 0, 0], [1, 2, 0], [1, 2, 3])
M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])
assert F == Fd
assert Fd == F
assert F != M
assert M != F
assert Fd != M
assert M != Fd
assert Fd == F.submodule(*F.basis())
assert Fd.is_full_module()
assert not M.is_full_module()
assert not Fd.is_zero()
assert not M.is_zero()
assert Fd.submodule().is_zero()
assert M.contains([x**2 + y**2 + x, 1 + y, 1])
assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
assert M.contains([y**2, 1 - x*y, -x])
assert not F.submodule([1 + x, 0, 0]) == F.submodule([1, 0, 0])
assert F.submodule([1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1])) == F
assert not M.is_submodule(0)
m = F.convert([x**2 + y**2, 1, 0])
n = M.convert(m)
assert m.module is F
assert n.module is M
raises(ValueError, lambda: M.submodule([1, 0, 0]))
raises(TypeError, lambda: M.union(1))
raises(ValueError, lambda: M.union(R.free_module(1).submodule([x])))
assert F.submodule([x, x, x]) != F.submodule([x, x, x], order="ilex")
def test_FreeModuleElement():
M = QQ.old_poly_ring(x).free_module(3)
e = M.convert([1, x, x**2])
f = [QQ.old_poly_ring(x).convert(1), QQ.old_poly_ring(x).convert(x), QQ.old_poly_ring(x).convert(x**2)]
assert list(e) == f
assert f[0] == e[0]
assert f[1] == e[1]
assert f[2] == e[2]
raises(IndexError, lambda: e[3])
g = M.convert([x, 0, 0])
assert e + g == M.convert([x + 1, x, x**2])
assert f + g == M.convert([x + 1, x, x**2])
assert -e == M.convert([-1, -x, -x**2])
assert e - g == M.convert([1 - x, x, x**2])
assert e != g
assert M.convert([x, x, x]) / QQ.old_poly_ring(x).convert(x) == [1, 1, 1]
R = QQ.old_poly_ring(x, order="ilex")
assert R.free_module(1).convert([x]) / R.convert(x) == [1]
def test_FreeModuleElement():
M = QQ.old_poly_ring(x).free_module(3)
e = M.convert([1, x, x**2])
f = [
QQ.old_poly_ring(x).convert(1),
QQ.old_poly_ring(x).convert(x),
QQ.old_poly_ring(x).convert(x**2)
]
assert list(e) == f
assert f[0] == e[0]
assert f[1] == e[1]
assert f[2] == e[2]
raises(IndexError, lambda: e[3])
g = M.convert([x, 0, 0])
assert e + g == M.convert([x + 1, x, x**2])
assert f + g == M.convert([x + 1, x, x**2])
assert -e == M.convert([-1, -x, -x**2])
assert e - g == M.convert([1 - x, x, x**2])
assert e != g
assert M.convert([x, x, x]) / QQ.old_poly_ring(x).convert(x) == [1, 1, 1]
R = QQ.old_poly_ring(x, order="ilex")
assert R.free_module(1).convert([x]) / R.convert(x) == [1]
def test_groebner_extendend():
M = QQ.old_poly_ring(x, y, z).free_module(3).submodule([x + 1, y, 1],
[x * y, z, z**2])
G, R = M._groebner_vec(extended=True)
for i, g in enumerate(G):
assert g == sum(c * gen for c, gen in zip(R[i], M.gens))
def test_groebner_extendend():
M = QQ.old_poly_ring(x, y, z).free_module(3).submodule([x + 1, y, 1], [x*y, z, z**2])
G, R = M._groebner_vec(extended=True)
for i, g in enumerate(G):
assert g == sum(c*gen for c, gen in zip(R[i], M.gens))
def test_quotient():
# SCA, example 1.8.13
R = QQ.old_poly_ring(x, y, z)
assert R.ideal(x, y).quotient(R.ideal(y ** 2, z)) == R.ideal(x, y)
def test_quotient():
# SCA, example 1.8.13
R = QQ.old_poly_ring(x, y, z)
assert R.ideal(x, y).quotient(R.ideal(y**2, z)) == R.ideal(x, y)
