def test_exceptions():
I = QQ.old_poly_ring(x).ideal(x)
J = QQ.old_poly_ring(y).ideal(1)
pytest.raises(ValueError, lambda: I.union(x))
pytest.raises(ValueError, lambda: I + J)
pytest.raises(ValueError, lambda: I * J)
pytest.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_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_reduction():
from diofant.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_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_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_PrettyPoly():
from diofant.polys.domains import QQ
F = QQ.frac_field(x, y)
R = QQ[x, y]
assert sstr(F.convert(x / (x + y))) == sstr(x / (x + y))
assert sstr(R.convert(x + y)) == sstr(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)