def test_QuotientRing():
I = QQ.old_poly_ring(x).ideal(x**2 + 1)
R = QQ.old_poly_ring(x) / I
assert R == QQ.old_poly_ring(x) / [x**2 + 1]
assert R == QQ.old_poly_ring(x) / QQ.old_poly_ring(x).ideal(x**2 + 1)
assert R != QQ.old_poly_ring(x)
assert R.convert(1) / x == -x + I
assert -1 + I == x**2 + I
assert R.convert(ZZ(1), ZZ) == 1 + I
assert R.convert(R.convert(x), R) == R.convert(x)
X = R.convert(x)
Y = QQ.old_poly_ring(x).convert(x)
assert -1 + I == X**2 + I
assert -1 + I == Y**2 + I
assert R.to_diofant(X) == x
pytest.raises(
ValueError,
lambda: QQ.old_poly_ring(x) / QQ.old_poly_ring(x, y).ideal(x))
R = QQ.old_poly_ring(x, order="ilex")
I = R.ideal(x)
assert R.convert(1) + I == (R / I).convert(1)
def test_operations():
F = QQ.old_poly_ring(x).free_module(2)
G = QQ.old_poly_ring(x).free_module(3)
f = F.identity_hom()
g = homomorphism(F, F, [0, [1, x]])
h = homomorphism(F, F, [[1, 0], 0])
i = homomorphism(F, G, [[1, 0, 0], [0, 1, 0]])
assert f == f
assert f != g
assert f != i
assert (f != F.identity_hom()) is False
assert 2 * f == f * 2 == homomorphism(F, F, [[2, 0], [0, 2]])
assert f / 2 == homomorphism(F, F,
[[Rational(1, 2), 0], [0, Rational(1, 2)]])
assert f + g == homomorphism(F, F, [[1, 0], [1, x + 1]])
assert f - g == homomorphism(F, F, [[1, 0], [-1, 1 - x]])
assert f * g == g == g * f
assert h * g == homomorphism(F, F, [0, [1, 0]])
assert g * h == homomorphism(F, F, [0, 0])
assert i * f == i
assert f([1, 2]) == [1, 2]
assert g([1, 2]) == [2, 2 * x]
assert i.restrict_domain(F.submodule([x, x]))([x, x]) == i([x, x])
h1 = h.quotient_domain(F.submodule([0, 1]))
assert h1([1, 0]) == h([1, 0])
assert h1.restrict_domain(h1.domain.submodule([x, 0]))([x, 0]) == h([x, 0])
pytest.raises(TypeError, lambda: f / g)
pytest.raises(TypeError, lambda: f + 1)
pytest.raises(TypeError, lambda: f + i)
pytest.raises(TypeError, lambda: f - 1)
pytest.raises(TypeError, lambda: f * i)
def test_properties():
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
h = homomorphism(F, F, [[x, 0], [y, 0]])
assert h.kernel() == F.submodule([-y, x])
assert h.image() == F.submodule([x, 0], [y, 0])
assert not h.is_injective()
assert not h.is_surjective()
assert h.restrict_codomain(h.image()).is_surjective()
assert h.restrict_domain(F.submodule([1, 0])).is_injective()
assert h.quotient_domain(h.kernel()).restrict_codomain(
h.image()).is_isomorphism()
R2 = QQ.old_poly_ring(x, y, order=(("lex", x), ("ilex", y))) / [x**2 + 1]
F = R2.free_module(2)
h = homomorphism(F, F, [[x, 0], [y, y + 1]])
assert h.is_isomorphism()
def test_creation():
F = QQ.old_poly_ring(x).free_module(3)
G = QQ.old_poly_ring(x).free_module(2)
SM = F.submodule([1, 1, 1])
Q = F / SM
SQ = Q.submodule([1, 0, 0])
matrix = [[1, 0], [0, 1], [-1, -1]]
h = homomorphism(F, G, matrix)
h2 = homomorphism(Q, G, matrix)
assert h.quotient_domain(SM) == h2
pytest.raises(ValueError,
lambda: h.quotient_domain(F.submodule([1, 0, 0])))
assert h2.restrict_domain(SQ) == homomorphism(SQ, G, matrix)
pytest.raises(ValueError, lambda: h.restrict_domain(G))
pytest.raises(ValueError, lambda: h.restrict_codomain(G.submodule([1, 0])))
pytest.raises(ValueError, lambda: h.quotient_codomain(F))
im = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
for M in [F, SM, Q, SQ]:
assert M.identity_hom() == homomorphism(M, M, im)
assert SM.inclusion_hom() == homomorphism(SM, F, im)
assert SQ.inclusion_hom() == homomorphism(SQ, Q, im)
assert Q.quotient_hom() == homomorphism(F, Q, im)
assert SQ.quotient_hom() == homomorphism(SQ.base, SQ, im)
class conv(object):
def convert(self, y=None):
return self
class dummy(object):
container = conv()
def submodule(*args):
return
pytest.raises(TypeError, lambda: homomorphism(dummy(), G, matrix))
pytest.raises(TypeError, lambda: homomorphism(F, dummy(), matrix))
pytest.raises(
ValueError,
lambda: homomorphism(QQ.old_poly_ring(x, y).free_module(3), G, matrix))
pytest.raises(ValueError, lambda: homomorphism(F, G, [0, 0]))
def test_printing():
R = QQ.old_poly_ring(x)
assert str(homomorphism(R.free_module(1), R.free_module(1), [0])) == \
'Matrix([[0]]) : QQ[x]**1 -> QQ[x]**1'
assert str(homomorphism(R.free_module(2), R.free_module(2), [0, 0])) == \
'Matrix([ \n[0, 0], : QQ[x]**2 -> QQ[x]**2\n[0, 0]]) '
assert str(homomorphism(R.free_module(1), R.free_module(1) / [[x]], [0])) == \
'Matrix([[0]]) : QQ[x]**1 -> QQ[x]**1/<[x]>'
assert str(R.free_module(
0).identity_hom()) == 'Matrix(0, 0, []) : QQ[x]**0 -> QQ[x]**0'
def test_QuotientRingElement():
R = QQ.old_poly_ring(x) / [x**10]
X = R.convert(x)
assert X * (X + 1) == R.convert(x**2 + x)
assert X * x == R.convert(x**2)
assert x * X == R.convert(x**2)
assert X + x == R.convert(2 * x)
assert x + X == 2 * X
assert X**2 == R.convert(x**2)
assert 1 / (1 - X) == R.convert(sum(x**i for i in range(10)))
assert X**10 == R.zero
assert X != x
pytest.raises(NotReversible, lambda: 1 / X)
