def dmp_ext_factor(f, u, K):
"""Factor multivariate polynomials over algebraic number fields. """
if not u:
return dup_ext_factor(f, K)
lc = dmp_ground_LC(f, u, K)
f = dmp_ground_monic(f, u, K)
if all(d <= 0 for d in dmp_degree_list(f, u)):
return lc, []
f, F = dmp_sqf_part(f, u, K), f
s, g, r = dmp_sqf_norm(f, u, K)
factors = dmp_factor_list_include(r, u, K.dom)
if len(factors) == 1:
coeff, factors = lc, [f]
else:
H = dmp_raise([K.one, s * K.unit], u, 0, K)
for i, (factor, _) in enumerate(factors):
h = dmp_convert(factor, u, K.dom, K)
h, _, g = dmp_inner_gcd(h, g, u, K)
h = dmp_compose(h, H, u, K)
factors[i] = h
return lc, dmp_trial_division(F, factors, u, K)
def dmp_ext_factor(f, u, K):
"""Factor multivariate polynomials over algebraic number fields. """
if not u:
return dup_ext_factor(f, K)
lc = dmp_ground_LC(f, u, K)
f = dmp_ground_monic(f, u, K)
if all([ d <= 0 for d in dmp_degree_list(f, u) ]):
return lc, []
f, F = dmp_sqf_part(f, u, K), f
s, g, r = dmp_sqf_norm(f, u, K)
factors = dmp_factor_list_include(r, u, K.dom)
if len(factors) == 1:
coeff, factors = lc, [f]
else:
H = dmp_raise([K.one, s*K.unit], u, 0, K)
for i, (factor, _) in enumerate(factors):
h = dmp_convert(factor, u, K.dom, K)
h, _, g = dmp_inner_gcd(h, g, u, K)
h = dmp_compose(h, H, u, K)
factors[i] = h
return lc, dmp_trial_division(F, factors, u, K)
def dmp_sqf_norm(f, u, K):
"""
Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.
Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> from sympy import I
>>> K = QQ.algebraic_field(I)
>>> R, x, y = ring("x,y", K)
>>> _, X, Y = ring("x,y", QQ)
>>> s, f, r = R.dmp_sqf_norm(x*y + y**2)
>>> s == 1
True
>>> f == x*y + y**2 + K([QQ(-1), QQ(0)])*y
True
>>> r == X**2*Y**2 + 2*X*Y**3 + Y**4 + Y**2
True
"""
if not u:
return dup_sqf_norm(f, K)
if not K.is_Algebraic:
raise DomainError("ground domain must be algebraic")
g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
F = dmp_raise([K.one, -K.unit], u, 0, K)
s = 0
while True:
h, _ = dmp_inject(f, u, K, front=True)
r = dmp_resultant(g, h, u + 1, K.dom)
if dmp_sqf_p(r, u, K.dom):
break
else:
f, s = dmp_compose(f, F, u, K), s + 1
return s, f, r
def dmp_sqf_norm(f, u, K):
"""
Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.
Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.
Examples
========
>>> from sympy.polys import ring, QQ
>>> from sympy import I
>>> K = QQ.algebraic_field(I)
>>> R, x, y = ring("x,y", K)
>>> _, X, Y = ring("x,y", QQ)
>>> s, f, r = R.dmp_sqf_norm(x*y + y**2)
>>> s == 1
True
>>> f == x*y + y**2 + K([QQ(-1), QQ(0)])*y
True
>>> r == X**2*Y**2 + 2*X*Y**3 + Y**4 + Y**2
True
"""
if not u:
return dup_sqf_norm(f, K)
if not K.is_Algebraic:
raise DomainError("ground domain must be algebraic")
g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
F = dmp_raise([K.one, -K.unit], u, 0, K)
s = 0
while True:
h, _ = dmp_inject(f, u, K, front=True)
r = dmp_resultant(g, h, u + 1, K.dom)
if dmp_sqf_p(r, u, K.dom):
break
else:
f, s = dmp_compose(f, F, u, K), s + 1
return s, f, r
def dmp_sqf_norm(f, u, K):
"""
Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.
Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.
Examples
========
>>> from sympy import I
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.sqfreetools import dmp_sqf_norm
>>> K = QQ.algebraic_field(I)
>>> s, f, r = dmp_sqf_norm([[K(1), K(0)], [K(1), K(0), K(0)]], 1, K)
>>> s == 1
True
>>> f == [[K(1), K(0)], [K(1), K([QQ(-1), QQ(0)]), K(0)]]
True
>>> r == [[1, 0, 0], [2, 0, 0, 0], [1, 0, 1, 0, 0]]
True
"""
if not u:
return dup_sqf_norm(f, K)
if not K.is_Algebraic:
raise DomainError("ground domain must be algebraic")
g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
F = dmp_raise([K.one, -K.unit], u, 0, K)
s = 0
while True:
h, _ = dmp_inject(f, u, K, front=True)
r = dmp_resultant(g, h, u + 1, K.dom)
if dmp_sqf_p(r, u, K.dom):
break
else:
f, s = dmp_compose(f, F, u, K), s + 1
return s, f, r
def dmp_sqf_norm(f, u, K):
"""
Square-free norm of ``f`` in ``K[X]``, useful over algebraic domains.
Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and ``r(x) = Norm(g(x))``
is a square-free polynomial over K, where ``a`` is the algebraic extension of ``K``.
Examples
========
>>> from sympy import I
>>> from sympy.polys.domains import QQ
>>> from sympy.polys.sqfreetools import dmp_sqf_norm
>>> K = QQ.algebraic_field(I)
>>> s, f, r = dmp_sqf_norm([[K(1), K(0)], [K(1), K(0), K(0)]], 1, K)
>>> s == 1
True
>>> f == [[K(1), K(0)], [K(1), K([QQ(-1), QQ(0)]), K(0)]]
True
>>> r == [[1, 0, 0], [2, 0, 0, 0], [1, 0, 1, 0, 0]]
True
"""
if not u:
return dup_sqf_norm(f, K)
if not K.is_Algebraic:
raise DomainError("ground domain must be algebraic")
g = dmp_raise(K.mod.rep, u + 1, 0, K.dom)
F = dmp_raise([K.one, -K.unit], u, 0, K)
s = 0
while True:
h, _ = dmp_inject(f, u, K, front=True)
r = dmp_resultant(g, h, u + 1, K.dom)
if dmp_sqf_p(r, u, K.dom):
break
else:
f, s = dmp_compose(f, F, u, K), s + 1
return s, f, r
def test_dmp_compose():
assert dmp_compose([1, 2, 1], [1, 2, 1], 0, ZZ) == [1, 4, 8, 8, 4]
assert dmp_compose([[[]]], [[[]]], 2, ZZ) == [[[]]]
assert dmp_compose([[[]]], [[[1]]], 2, ZZ) == [[[]]]
assert dmp_compose([[[]]], [[[1]], [[2]]], 2, ZZ) == [[[]]]
assert dmp_compose([[[1]]], [], 2, ZZ) == [[[1]]]
assert dmp_compose([[1], [2], [ ]], [[]], 1, ZZ) == [[]]
assert dmp_compose([[1], [2], [1]], [[]], 1, ZZ) == [[1]]
assert dmp_compose([[1], [2], [1]], [[1]], 1, ZZ) == [[4]]
assert dmp_compose([[1], [2], [1]], [[7]], 1, ZZ) == [[64]]
assert dmp_compose([[1], [2], [1]], [[1], [-1]], 1, ZZ) == [[1], [ ], [ ]]
assert dmp_compose([[1], [2], [1]], [[1], [ 1]], 1, ZZ) == [[1], [4], [4]]
assert dmp_compose(
[[1], [2], [1]], [[1], [2], [1]], 1, ZZ) == [[1], [4], [8], [8], [4]]
def compose(f, g):
"""Computes functional composition of `f` and `g`. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_compose(F, G, lev, dom))
def test_dmp_compose():
assert dmp_compose([1, 2, 1], [1, 2, 1], 0, ZZ) == [1, 4, 8, 8, 4]
assert dmp_compose([[[]]], [[[]]], 2, ZZ) == [[[]]]
assert dmp_compose([[[]]], [[[1]]], 2, ZZ) == [[[]]]
assert dmp_compose([[[]]], [[[1]], [[2]]], 2, ZZ) == [[[]]]
assert dmp_compose([[[1]]], [], 2, ZZ) == [[[1]]]
assert dmp_compose([[1], [2], []], [[]], 1, ZZ) == [[]]
assert dmp_compose([[1], [2], [1]], [[]], 1, ZZ) == [[1]]
assert dmp_compose([[1], [2], [1]], [[1]], 1, ZZ) == [[4]]
assert dmp_compose([[1], [2], [1]], [[7]], 1, ZZ) == [[64]]
assert dmp_compose([[1], [2], [1]], [[1], [-1]], 1, ZZ) == [[1], [], []]
assert dmp_compose([[1], [2], [1]], [[1], [1]], 1, ZZ) == [[1], [4], [4]]
assert dmp_compose([[1], [2], [1]], [[1], [2], [1]], 1, ZZ) == [[1], [4],
[8], [8],
[4]]
def compose(f, g):
"""Computes functional composition of `f` and `g`. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_compose(F, G, lev, dom))
