def test_FreeModule():
M1 = FreeModule(QQ[x], 2)
assert M1 == FreeModule(QQ[x], 2)
assert M1 != FreeModule(QQ[y], 2)
assert M1 != FreeModule(QQ[x], 3)
M2 = FreeModule(QQ.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[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.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[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[x].free_module(2))
raises(NotImplementedError, lambda: FreeModulePolyRing(ZZ, 2))
raises(CoercionFailed, lambda: M1.convert(QQ[x].free_module(3)
.convert([1, 2, 3])))
raises(CoercionFailed, lambda: M3.convert(1))
def test_FreeModule():
M1 = FreeModule(QQ[x], 2)
assert M1 == FreeModule(QQ[x], 2)
assert M1 != FreeModule(QQ[y], 2)
assert M1 != FreeModule(QQ[x], 3)
M2 = FreeModule(QQ.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[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.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[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[x].free_module(2))
raises(NotImplementedError, lambda: FreeModulePolyRing(ZZ, 2))
raises(CoercionFailed,
lambda: M1.convert(QQ[x].free_module(3).convert([1, 2, 3])))
raises(CoercionFailed, lambda: M3.convert(1))
def test_SubModulePolyRing_local():
R = QQ.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.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_intersection():
R = QQ[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.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_intersection():
R = QQ[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.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_nontriv_local():
R = QQ.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_in_terms_of_generators():
R = QQ.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)]
R = QQ[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_SubModulePolyRing_nontriv_local():
R = QQ.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_nontriv_local():
R = QQ.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.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_in_terms_of_generators():
R = QQ.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[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_FreeModuleElement():
M = QQ[x].free_module(3)
e = M.convert([1, x, x**2])
f = [QQ[x].convert(1), QQ[x].convert(x), QQ[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[x].convert(x) == [1, 1, 1]
R = QQ.poly_ring(x, order="ilex")
assert R.free_module(1).convert([x]) / R.convert(x) == [1]
def test_FreeModuleElement():
M = QQ[x].free_module(3)
e = M.convert([1, x, x**2])
f = [QQ[x].convert(1), QQ[x].convert(x), QQ[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[x].convert(x) == [1, 1, 1]
R = QQ.poly_ring(x, order="ilex")
assert R.free_module(1).convert([x]) / R.convert(x) == [1]
def test_ModulesQuotientRing():
R = QQ.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[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.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[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)
