132 lines
3.7 KiB
Python
132 lines
3.7 KiB
Python
|
"""Test ideals.py code."""
|
||
|
|
|
||
|
|
from sympy.polys import QQ, ilex
|
||
|
|
from sympy.abc import x, y, z
|
||
|
|
from sympy.testing.pytest import raises
|
||
|
|
|
||
|
|
|
||
|
|
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_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_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_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_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 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_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)
|