def test_localring():
Qxy = QQ.old_frac_field(x, y)
R = QQ.old_poly_ring(x, y, order="ilex")
X = R.convert(x)
Y = R.convert(y)
assert x in R
assert 1/x not in R
assert 1/(1 + x) in R
assert Y in R
assert X.ring == R
assert X*(Y**2 + 1)/(1 + X) == R.convert(x*(y**2 + 1)/(1 + x))
assert X*y == X*Y
raises(ExactQuotientFailed, lambda: X/Y)
raises(ExactQuotientFailed, lambda: x/Y)
raises(ExactQuotientFailed, lambda: X/y)
assert X + y == X + Y == R.convert(x + y) == x + Y
assert X - y == X - Y == R.convert(x - y) == x - Y
assert X + 1 == R.convert(x + 1)
assert X**2 / X == X
assert R.from_GlobalPolynomialRing(ZZ.old_poly_ring(x, y).convert(x), ZZ.old_poly_ring(x, y)) == X
assert R.from_FractionField(Qxy.convert(x), Qxy) == X
raises(CoercionFailed, lambda: R.from_FractionField(Qxy.convert(x)/y, Qxy))
raises(ExactQuotientFailed, lambda: X/Y)
raises(NotReversible, lambda: X.invert())
assert R._sdm_to_vector(
R._vector_to_sdm([X/(X + 1), Y/(1 + X*Y)], R.order), 2) == \
[X*(1 + X*Y), Y*(1 + X)]
def test_globalring():
Qxy = QQ.old_frac_field(x, y)
R = QQ.old_poly_ring(x, y)
X = R.convert(x)
Y = R.convert(y)
assert x in R
assert 1/x not in R
assert 1/(1 + x) not in R
assert Y in R
assert X.ring == R
assert X * (Y**2 + 1) == R.convert(x * (y**2 + 1))
assert X * y == X * Y == R.convert(x * y) == x * Y
assert X + y == X + Y == R.convert(x + y) == x + Y
assert X - y == X - Y == R.convert(x - y) == x - Y
assert X + 1 == R.convert(x + 1)
raises(ExactQuotientFailed, lambda: X/Y)
raises(ExactQuotientFailed, lambda: x/Y)
raises(ExactQuotientFailed, lambda: X/y)
assert X**2 / X == X
assert R.from_GlobalPolynomialRing(ZZ.old_poly_ring(x, y).convert(x), ZZ.old_poly_ring(x, y)) == X
assert R.from_FractionField(Qxy.convert(x), Qxy) == X
assert R.from_FractionField(Qxy.convert(x)/y, Qxy) is None
assert R._sdm_to_vector(R._vector_to_sdm([X, Y], R.order), 2) == [X, Y]
def test_localring():
Qxy = QQ.old_frac_field(x, y)
R = QQ.old_poly_ring(x, y, order="ilex")
X = R.convert(x)
Y = R.convert(y)
assert x in R
assert 1/x not in R
assert 1/(1 + x) in R
assert Y in R
assert X.ring == R
assert X*(Y**2 + 1)/(1 + X) == R.convert(x*(y**2 + 1)/(1 + x))
assert X*y == X*Y
raises(ExactQuotientFailed, lambda: X/Y)
raises(ExactQuotientFailed, lambda: x/Y)
raises(ExactQuotientFailed, lambda: X/y)
assert X + y == X + Y == R.convert(x + y) == x + Y
assert X - y == X - Y == R.convert(x - y) == x - Y
assert X + 1 == R.convert(x + 1)
assert X**2 / X == X
assert R.from_GlobalPolynomialRing(ZZ.old_poly_ring(x, y).convert(x), ZZ.old_poly_ring(x, y)) == X
assert R.from_FractionField(Qxy.convert(x), Qxy) == X
raises(CoercionFailed, lambda: R.from_FractionField(Qxy.convert(x)/y, Qxy))
raises(ExactQuotientFailed, lambda: X/Y)
raises(NotReversible, lambda: X.invert())
assert R._sdm_to_vector(
R._vector_to_sdm([X/(X + 1), Y/(1 + X*Y)], R.order), 2) == \
[X*(1 + X*Y), Y*(1 + X)]
def test_globalring():
Qxy = QQ.old_frac_field(x, y)
R = QQ.old_poly_ring(x, y)
X = R.convert(x)
Y = R.convert(y)
assert x in R
assert 1/x not in R
assert 1/(1 + x) not in R
assert Y in R
assert X.ring == R
assert X * (Y**2 + 1) == R.convert(x * (y**2 + 1))
assert X * y == X * Y == R.convert(x * y) == x * Y
assert X + y == X + Y == R.convert(x + y) == x + Y
assert X - y == X - Y == R.convert(x - y) == x - Y
assert X + 1 == R.convert(x + 1)
raises(ExactQuotientFailed, lambda: X/Y)
raises(ExactQuotientFailed, lambda: x/Y)
raises(ExactQuotientFailed, lambda: X/y)
assert X**2 / X == X
assert R.from_GlobalPolynomialRing(ZZ.old_poly_ring(x, y).convert(x), ZZ.old_poly_ring(x, y)) == X
assert R.from_FractionField(Qxy.convert(x), Qxy) == X
assert R.from_FractionField(Qxy.convert(x)/y, Qxy) is None
assert R._sdm_to_vector(R._vector_to_sdm([X, Y], R.order), 2) == [X, Y]
def _normalize(list_of, parent, negative=True):
"""
Normalize a given annihilator
"""
num = []
denom = []
base = parent.base
K = base.get_field()
R = ZZ.old_poly_ring(base.gens[0])
lcm_denom = R.from_sympy(S(1))
list_of_coeff = []
# convert polynomials to the elements of associated
# fraction field
for i, j in enumerate(list_of):
if isinstance(j, base.dtype):
list_of_coeff.append(K.new(j.rep))
elif not isinstance(j, K.dtype):
list_of_coeff.append(K.from_sympy(sympify(j)))
else:
list_of_coeff.append(j)
# corresponding numerators of the sequence of polynomials
num.append(base(list_of_coeff[i].num))
# corresponding denominators
den = list_of_coeff[i].den
if isinstance(den[0], PythonRational):
for i, j in enumerate(den):
den[i] = j.p
denom.append(R(den))
# lcm of denominators in the coefficients
for i in denom:
lcm_denom = i.lcm(lcm_denom)
if negative is True:
lcm_denom = -lcm_denom
lcm_denom = K.new(lcm_denom.rep)
# multiply the coefficients with lcm
for i, j in enumerate(list_of_coeff):
list_of_coeff[i] = j * lcm_denom
gcd_numer = base.from_FractionField(list_of_coeff[-1], K)
# gcd of numerators in the coefficients
for i in num:
gcd_numer = i.gcd(gcd_numer)
gcd_numer = K.new(gcd_numer.rep)
# divide all the coefficients by the gcd
for i, j in enumerate(list_of_coeff):
list_of_coeff[i] = base.from_FractionField(j / gcd_numer, K)
return DifferentialOperator(list_of_coeff, parent)
def test_GlobalPolynomialRing_convert():
K1 = QQ.old_poly_ring(x)
K2 = QQ[x]
assert K1.convert(x) == K1.convert(K2.convert(x), K2)
assert K2.convert(x) == K2.convert(K1.convert(x), K1)
K1 = QQ.old_poly_ring(x, y)
K2 = QQ[x]
assert K1.convert(x) == K1.convert(K2.convert(x), K2)
#assert K2.convert(x) == K2.convert(K1.convert(x), K1)
K1 = ZZ.old_poly_ring(x, y)
K2 = QQ[x]
assert K1.convert(x) == K1.convert(K2.convert(x), K2)
def test_units():
R = QQ.old_poly_ring(x)
assert R.is_unit(R.convert(1))
assert R.is_unit(R.convert(2))
assert not R.is_unit(R.convert(x))
assert not R.is_unit(R.convert(1 + x))
R = QQ.old_poly_ring(x, order='ilex')
assert R.is_unit(R.convert(1))
assert R.is_unit(R.convert(2))
assert not R.is_unit(R.convert(x))
assert R.is_unit(R.convert(1 + x))
R = ZZ.old_poly_ring(x)
assert R.is_unit(R.convert(1))
assert not R.is_unit(R.convert(2))
assert not R.is_unit(R.convert(x))
assert not R.is_unit(R.convert(1 + x))
def test_units():
R = QQ.old_poly_ring(x)
assert R.is_unit(R.convert(1))
assert R.is_unit(R.convert(2))
assert not R.is_unit(R.convert(x))
assert not R.is_unit(R.convert(1 + x))
R = QQ.old_poly_ring(x, order='ilex')
assert R.is_unit(R.convert(1))
assert R.is_unit(R.convert(2))
assert not R.is_unit(R.convert(x))
assert R.is_unit(R.convert(1 + x))
R = ZZ.old_poly_ring(x)
assert R.is_unit(R.convert(1))
assert not R.is_unit(R.convert(2))
assert not R.is_unit(R.convert(x))
assert not R.is_unit(R.convert(1 + x))
def _normalize(list_of, parent, negative=True):
"""
Normalize a given annihilator
"""
num = []
denom = []
base = parent.base
K = base.get_field()
R = ZZ.old_poly_ring(base.gens[0])
lcm_denom = R.from_sympy(S(1))
list_of_coeff = []
for i, j in enumerate(list_of):
if not isinstance(j, K.dtype):
list_of_coeff.append(K.from_sympy(sympify(j)))
else:
list_of_coeff.append(j)
num.append(base(list_of_coeff[i].num))
den = list_of_coeff[i].den
if isinstance(den[0], PythonRational):
for i, j in enumerate(den):
den[i] = j.p
denom.append(R(den))
for i in denom:
lcm_denom = i.lcm(lcm_denom)
if negative is True:
lcm_denom = -lcm_denom
lcm_denom = K.new(lcm_denom.rep)
for i, j in enumerate(list_of_coeff):
list_of_coeff[i] = j * lcm_denom
gcd_numer = base.from_FractionField(list_of_coeff[-1], K)
for i in num:
gcd_numer = i.gcd(gcd_numer)
gcd_numer = K.new(gcd_numer.rep)
for i, j in enumerate(list_of_coeff):
list_of_coeff[i] = base.from_FractionField(j / gcd_numer, K)
return DifferentialOperator(list_of_coeff, parent)
def _normalize(list_of, parent, negative=True):
"""
Normalize a given annihilator
"""
num = []
denom = []
base = parent.base
K = base.get_field()
R = ZZ.old_poly_ring(base.gens[0])
lcm_denom = R.from_sympy(S(1))
list_of_coeff = []
for i, j in enumerate(list_of):
if not isinstance(j, K.dtype):
list_of_coeff.append(K.from_sympy(sympify(j)))
else:
list_of_coeff.append(j)
num.append(base(list_of_coeff[i].num))
den = list_of_coeff[i].den
if isinstance(den[0], PythonRational):
for i, j in enumerate(den):
den[i] = j.p
denom.append(R(den))
for i in denom:
lcm_denom = i.lcm(lcm_denom)
if negative is True:
lcm_denom = -lcm_denom
lcm_denom = K.new(lcm_denom.rep)
for i, j in enumerate(list_of_coeff):
list_of_coeff[i] = j * lcm_denom
gcd_numer = base.from_FractionField(list_of_coeff[-1], K)
for i in num:
gcd_numer = i.gcd(gcd_numer)
gcd_numer = K.new(gcd_numer.rep)
for i, j in enumerate(list_of_coeff):
list_of_coeff[i] = base.from_FractionField(j / gcd_numer, K)
return DifferentialOperator(list_of_coeff, parent)