def dup_sqf_part(f, K):
"""
Returns square-free part of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dup_sqf_part
>>> dup_sqf_part([ZZ(1), ZZ(0), -ZZ(3), -ZZ(2)], ZZ)
[1, -1, -2]
"""
if not K.has_CharacteristicZero:
return dup_gf_sqf_part(f, K)
if not f:
return f
if K.is_negative(dup_LC(f, K)):
f = dup_neg(f, K)
gcd = dup_gcd(f, dup_diff(f, 1, K), K)
sqf = dup_quo(f, gcd, K)
if K.has_Field or not K.is_Exact:
return dup_monic(sqf, K)
else:
return dup_primitive(sqf, K)[1]
def dup_gff_list(f, K):
"""
Compute greatest factorial factorization of ``f`` in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_gff_list(x**5 + 2*x**4 - x**3 - 2*x**2)
[(x, 1), (x + 2, 4)]
"""
if not f:
raise ValueError("greatest factorial factorization doesn't exist for a zero polynomial")
f = dup_monic(f, K)
if not dup_degree(f):
return []
else:
g = dup_gcd(f, dup_shift(f, K.one, K), K)
H = dup_gff_list(g, K)
for i, (h, k) in enumerate(H):
g = dup_mul(g, dup_shift(h, -K(k), K), K)
H[i] = (h, k + 1)
f = dup_quo(f, g, K)
if not dup_degree(f):
return H
else:
return [(f, 1)] + H
def dup_sqf_part(f, K):
"""
Returns square-free part of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_sqf_part(x**3 - 3*x - 2)
x**2 - x - 2
"""
if K.is_FiniteField:
return dup_gf_sqf_part(f, K)
if not f:
return f
if K.is_negative(dup_LC(f, K)):
f = dup_neg(f, K)
gcd = dup_gcd(f, dup_diff(f, 1, K), K)
sqf = dup_quo(f, gcd, K)
if K.has_Field:
return dup_monic(sqf, K)
else:
return dup_primitive(sqf, K)[1]
def dup_sqf_part(f, K):
"""
Returns square-free part of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_sqf_part(x**3 - 3*x - 2)
x**2 - x - 2
"""
if K.is_FiniteField:
return dup_gf_sqf_part(f, K)
if not f:
return f
if K.is_negative(dup_LC(f, K)):
f = dup_neg(f, K)
gcd = dup_gcd(f, dup_diff(f, 1, K), K)
sqf = dup_quo(f, gcd, K)
if K.is_Field:
return dup_monic(sqf, K)
else:
return dup_primitive(sqf, K)[1]
def dup_gff_list(f, K):
"""
Compute greatest factorial factorization of ``f`` in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_gff_list(x**5 + 2*x**4 - x**3 - 2*x**2)
[(x, 1), (x + 2, 4)]
"""
if not f:
raise ValueError("greatest factorial factorization doesn't exist for a zero polynomial")
f = dup_monic(f, K)
if not dup_degree(f):
return []
else:
g = dup_gcd(f, dup_shift(f, K.one, K), K)
H = dup_gff_list(g, K)
for i, (h, k) in enumerate(H):
g = dup_mul(g, dup_shift(h, -K(k), K), K)
H[i] = (h, k + 1)
f = dup_quo(f, g, K)
if not dup_degree(f):
return H
else:
return [(f, 1)] + H
def dup_sqf_part(f, K):
"""
Returns square-free part of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dup_sqf_part
>>> dup_sqf_part([ZZ(1), ZZ(0), -ZZ(3), -ZZ(2)], ZZ)
[1, -1, -2]
"""
if not K.has_CharacteristicZero:
return dup_gf_sqf_part(f, K)
if not f:
return f
if K.is_negative(dup_LC(f, K)):
f = dup_neg(f, K)
gcd = dup_gcd(f, dup_diff(f, 1, K), K)
sqf = dup_quo(f, gcd, K)
if K.has_Field or not K.is_Exact:
return dup_monic(sqf, K)
else:
return dup_primitive(sqf, K)[1]
def dup_sqf_p(f, K):
"""
Return ``True`` if ``f`` is a square-free polynomial in ``K[x]``.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dup_sqf_p
>>> dup_sqf_p([ZZ(1),-ZZ(2), ZZ(1)], ZZ)
False
>>> dup_sqf_p([ZZ(1), ZZ(0),-ZZ(1)], ZZ)
True
"""
if not f:
return True
else:
return not dup_degree(dup_gcd(f, dup_diff(f, 1, K), K))
def dup_sqf_p(f, K):
"""
Return ``True`` if ``f`` is a square-free polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_sqf_p(x**2 - 2*x + 1)
False
>>> R.dup_sqf_p(x**2 - 1)
True
"""
if not f:
return True
else:
return not dup_degree(dup_gcd(f, dup_diff(f, 1, K), K))
def dup_sqf_p(f, K):
"""
Return ``True`` if ``f`` is a square-free polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_sqf_p(x**2 - 2*x + 1)
False
>>> R.dup_sqf_p(x**2 - 1)
True
"""
if not f:
return True
else:
return not dup_degree(dup_gcd(f, dup_diff(f, 1, K), K))
def dup_gff_list(f, K):
"""
Compute greatest factorial factorization of ``f`` in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dup_gff_list
>>> f = ZZ.map([1, 2, -1, -2, 0, 0])
>>> dup_gff_list(f, ZZ)
[([1, 0], 1), ([1, 2], 4)]
"""
if not f:
raise ValueError(
"greatest factorial factorization doesn't exist for a zero polynomial"
)
f = dup_monic(f, K)
if not dup_degree(f):
return []
else:
g = dup_gcd(f, dup_shift(f, K.one, K), K)
H = dup_gff_list(g, K)
for i, (h, k) in enumerate(H):
g = dup_mul(g, dup_shift(h, -K(k), K), K)
H[i] = (h, k + 1)
f = dup_quo(f, g, K)
if not dup_degree(f):
return H
else:
return [(f, 1)] + H
def dup_gff_list(f, K):
"""
Compute greatest factorial factorization of ``f`` in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dup_gff_list
>>> f = ZZ.map([1, 2, -1, -2, 0, 0])
>>> dup_gff_list(f, ZZ)
[([1, 0], 1), ([1, 2], 4)]
"""
if not f:
raise ValueError("greatest factorial factorization doesn't exist for a zero polynomial")
f = dup_monic(f, K)
if not dup_degree(f):
return []
else:
g = dup_gcd(f, dup_shift(f, K.one, K), K)
H = dup_gff_list(g, K)
for i, (h, k) in enumerate(H):
g = dup_mul(g, dup_shift(h, -K(k), K), K)
H[i] = (h, k + 1)
f = dup_quo(f, g, K)
if not dup_degree(f):
return H
else:
return [(f, 1)] + H