def dup_sqf_list(f, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dup_sqf_list
>>> f = ZZ.map([2, 16, 50, 76, 56, 16])
>>> dup_sqf_list(f, ZZ)
(2, [([1, 1], 2), ([1, 2], 3)])
>>> dup_sqf_list(f, ZZ, all=True)
(2, [([1], 1), ([1, 1], 2), ([1, 2], 3)])
"""
if not K.has_CharacteristicZero:
return dup_gf_sqf_list(f, K, all=all)
if K.has_Field or not K.is_Exact:
coeff = dup_LC(f, K)
f = dup_monic(f, K)
else:
coeff, f = dup_primitive(f, K)
if K.is_negative(dup_LC(f, K)):
f = dup_neg(f, K)
coeff = -coeff
if dup_degree(f) <= 0:
return coeff, []
result, i = [], 1
h = dup_diff(f, 1, K)
g, p, q = dup_inner_gcd(f, h, K)
while True:
d = dup_diff(p, 1, K)
h = dup_sub(q, d, K)
if not h:
result.append((p, i))
break
g, p, q = dup_inner_gcd(p, h, K)
if all or dup_degree(g) > 0:
result.append((g, i))
i += 1
return coeff, result
def dup_sqf_list(f, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.sqfreetools import dup_sqf_list
>>> f = ZZ.map([2, 16, 50, 76, 56, 16])
>>> dup_sqf_list(f, ZZ)
(2, [([1, 1], 2), ([1, 2], 3)])
>>> dup_sqf_list(f, ZZ, all=True)
(2, [([1], 1), ([1, 1], 2), ([1, 2], 3)])
"""
if not K.has_CharacteristicZero:
return dup_gf_sqf_list(f, K, all=all)
if K.has_Field or not K.is_Exact:
coeff = dup_LC(f, K)
f = dup_monic(f, K)
else:
coeff, f = dup_primitive(f, K)
if K.is_negative(dup_LC(f, K)):
f = dup_neg(f, K)
coeff = -coeff
if dup_degree(f) <= 0:
return coeff, []
result, i = [], 1
h = dup_diff(f, 1, K)
g, p, q = dup_inner_gcd(f, h, K)
while True:
d = dup_diff(p, 1, K)
h = dup_sub(q, d, K)
if not h:
result.append((p, i))
break
g, p, q = dup_inner_gcd(p, h, K)
if all or dup_degree(g) > 0:
result.append((g, i))
i += 1
return coeff, result
def dup_sqf_list(f, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16
>>> R.dup_sqf_list(f)
(2, [(x + 1, 2), (x + 2, 3)])
>>> R.dup_sqf_list(f, all=True)
(2, [(1, 1), (x + 1, 2), (x + 2, 3)])
"""
if K.is_FiniteField:
return dup_gf_sqf_list(f, K, all=all)
if K.has_Field:
coeff = dup_LC(f, K)
f = dup_monic(f, K)
else:
coeff, f = dup_primitive(f, K)
if K.is_negative(dup_LC(f, K)):
f = dup_neg(f, K)
coeff = -coeff
if dup_degree(f) <= 0:
return coeff, []
result, i = [], 1
h = dup_diff(f, 1, K)
g, p, q = dup_inner_gcd(f, h, K)
while True:
d = dup_diff(p, 1, K)
h = dup_sub(q, d, K)
if not h:
result.append((p, i))
break
g, p, q = dup_inner_gcd(p, h, K)
if all or dup_degree(g) > 0:
result.append((g, i))
i += 1
return coeff, result
def dup_sqf_list(f, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[x]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16
>>> R.dup_sqf_list(f)
(2, [(x + 1, 2), (x + 2, 3)])
>>> R.dup_sqf_list(f, all=True)
(2, [(1, 1), (x + 1, 2), (x + 2, 3)])
"""
if K.is_FiniteField:
return dup_gf_sqf_list(f, K, all=all)
if K.is_Field:
coeff = dup_LC(f, K)
f = dup_monic(f, K)
else:
coeff, f = dup_primitive(f, K)
if K.is_negative(dup_LC(f, K)):
f = dup_neg(f, K)
coeff = -coeff
if dup_degree(f) <= 0:
return coeff, []
result, i = [], 1
h = dup_diff(f, 1, K)
g, p, q = dup_inner_gcd(f, h, K)
while True:
d = dup_diff(p, 1, K)
h = dup_sub(q, d, K)
if not h:
result.append((p, i))
break
g, p, q = dup_inner_gcd(p, h, K)
if all or dup_degree(g) > 0:
result.append((g, i))
i += 1
return coeff, result
def test_dmp_diff():
assert dmp_diff([], 1, 0, ZZ) == []
assert dmp_diff([[]], 1, 1, ZZ) == [[]]
assert dmp_diff([[[]]], 1, 2, ZZ) == [[[]]]
assert dmp_diff([[[1], [2]]], 1, 2, ZZ) == [[[]]]
assert dmp_diff([[[1]], [[]]], 1, 2, ZZ) == [[[1]]]
assert dmp_diff([[[3]], [[1]], [[]]], 1, 2, ZZ) == [[[6]], [[1]]]
assert dmp_diff([1, -1, 0, 0, 2], 1, 0, ZZ) == \
dup_diff([1, -1, 0, 0, 2], 1, ZZ)
assert dmp_diff(f_6, 0, 3, ZZ) == f_6
assert dmp_diff(f_6, 1, 3, ZZ) == [[[[8460]], [[]]],
[[[135, 0, 0], [], [], [-135, 0, 0]]],
[[[]]],
[[[-423]], [[-47]], [[]],
[[141], [], [94, 0], []], [[]]]]
assert dmp_diff(f_6, 2, 3, ZZ) == dmp_diff(dmp_diff(f_6, 1, 3, ZZ), 1, 3,
ZZ)
assert dmp_diff(f_6, 3, 3, ZZ) == dmp_diff(
dmp_diff(dmp_diff(f_6, 1, 3, ZZ), 1, 3, ZZ), 1, 3, ZZ)
K = FF(23)
F_6 = dmp_normal(f_6, 3, K)
assert dmp_diff(F_6, 0, 3, K) == F_6
assert dmp_diff(F_6, 1, 3, K) == dmp_diff(F_6, 1, 3, K)
assert dmp_diff(F_6, 2, 3, K) == dmp_diff(dmp_diff(F_6, 1, 3, K), 1, 3, K)
assert dmp_diff(F_6, 3, 3,
K) == dmp_diff(dmp_diff(dmp_diff(F_6, 1, 3, K), 1, 3, K),
1, 3, K)
def test_dmp_diff():
assert dmp_diff([], 1, 0, ZZ) == []
assert dmp_diff([[]], 1, 1, ZZ) == [[]]
assert dmp_diff([[[]]], 1, 2, ZZ) == [[[]]]
assert dmp_diff([[[1], [2]]], 1, 2, ZZ) == [[[]]]
assert dmp_diff([[[1]], [[]]], 1, 2, ZZ) == [[[1]]]
assert dmp_diff([[[3]], [[1]], [[]]], 1, 2, ZZ) == [[[6]], [[1]]]
assert dmp_diff([1, -1, 0, 0, 2], 1, 0, ZZ) == \
dup_diff([1, -1, 0, 0, 2], 1, ZZ)
assert dmp_diff(f_6, 0, 3, ZZ) == f_6
assert dmp_diff(f_6, 1, 3, ZZ) == dmp_diff(f_6, 1, 3, ZZ)
assert dmp_diff(
f_6, 2, 3, ZZ) == dmp_diff(dmp_diff(f_6, 1, 3, ZZ), 1, 3, ZZ)
assert dmp_diff(f_6, 3, 3, ZZ) == dmp_diff(
dmp_diff(dmp_diff(f_6, 1, 3, ZZ), 1, 3, ZZ), 1, 3, ZZ)
K = FF(23)
F_6 = dmp_normal(f_6, 3, K)
assert dmp_diff(F_6, 0, 3, K) == F_6
assert dmp_diff(F_6, 1, 3, K) == dmp_diff(F_6, 1, 3, K)
assert dmp_diff(F_6, 2, 3, K) == dmp_diff(dmp_diff(F_6, 1, 3, K), 1, 3, K)
assert dmp_diff(F_6, 3, 3, K) == dmp_diff(
dmp_diff(dmp_diff(F_6, 1, 3, K), 1, 3, K), 1, 3, K)
def dup_discriminant(f, K):
"""
Computes discriminant of a polynomial in `K[x]`.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_discriminant
>>> dup_discriminant([ZZ(1), ZZ(2), ZZ(3)], ZZ)
-8
"""
d = dup_degree(f)
if d <= 0:
return K.zero
else:
s = (-1)**((d * (d - 1)) // 2)
c = dup_LC(f, K)
r = dup_resultant(f, dup_diff(f, 1, K), K)
return K.quo(r, c * K(s))
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_discriminant(f, K):
"""
Computes discriminant of a polynomial in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_discriminant(x**2 + 2*x + 3)
-8
"""
d = dup_degree(f)
if d <= 0:
return K.zero
else:
s = (-1)**((d*(d - 1)) // 2)
c = dup_LC(f, K)
r = dup_resultant(f, dup_diff(f, 1, K), K)
return K.quo(r, c*K(s))
def dup_discriminant(f, K):
"""
Computes discriminant of a polynomial in `K[x]`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_discriminant(x**2 + 2*x + 3)
-8
"""
d = dup_degree(f)
if d <= 0:
return K.zero
else:
s = (-1)**((d*(d - 1)) // 2)
c = dup_LC(f, K)
r = dup_resultant(f, dup_diff(f, 1, K), K)
return K.quo(r, c*K(s))
def dup_discriminant(f, K):
"""
Computes discriminant of a polynomial in `K[x]`.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dup_discriminant
>>> dup_discriminant([ZZ(1), ZZ(2), ZZ(3)], ZZ)
-8
"""
d = dup_degree(f)
if d <= 0:
return K.zero
else:
s = (-1)**((d*(d-1)) // 2)
c = dup_LC(f, K)
r = dup_resultant(f, dup_diff(f, 1, K), K)
return K.quo(r, c*K(s))
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_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_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_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 test_dup_gcd():
assert dup_zz_heu_gcd([], [], ZZ) == ([], [], [])
assert dup_rr_prs_gcd([], [], ZZ) == ([], [], [])
assert dup_zz_heu_gcd([2], [], ZZ) == ([2], [1], [])
assert dup_rr_prs_gcd([2], [], ZZ) == ([2], [1], [])
assert dup_zz_heu_gcd([-2], [], ZZ) == ([2], [-1], [])
assert dup_rr_prs_gcd([-2], [], ZZ) == ([2], [-1], [])
assert dup_zz_heu_gcd([], [-2], ZZ) == ([2], [], [-1])
assert dup_rr_prs_gcd([], [-2], ZZ) == ([2], [], [-1])
assert dup_zz_heu_gcd([], [2, 4], ZZ) == ([2, 4], [], [1])
assert dup_rr_prs_gcd([], [2, 4], ZZ) == ([2, 4], [], [1])
assert dup_zz_heu_gcd([2, 4], [], ZZ) == ([2, 4], [1], [])
assert dup_rr_prs_gcd([2, 4], [], ZZ) == ([2, 4], [1], [])
assert dup_zz_heu_gcd([2], [2], ZZ) == ([2], [1], [1])
assert dup_rr_prs_gcd([2], [2], ZZ) == ([2], [1], [1])
assert dup_zz_heu_gcd([-2], [2], ZZ) == ([2], [-1], [1])
assert dup_rr_prs_gcd([-2], [2], ZZ) == ([2], [-1], [1])
assert dup_zz_heu_gcd([2], [-2], ZZ) == ([2], [1], [-1])
assert dup_rr_prs_gcd([2], [-2], ZZ) == ([2], [1], [-1])
assert dup_zz_heu_gcd([-2], [-2], ZZ) == ([2], [-1], [-1])
assert dup_rr_prs_gcd([-2], [-2], ZZ) == ([2], [-1], [-1])
assert dup_zz_heu_gcd([1, 2, 1], [1], ZZ) == ([1], [1, 2, 1], [1])
assert dup_rr_prs_gcd([1, 2, 1], [1], ZZ) == ([1], [1, 2, 1], [1])
assert dup_zz_heu_gcd([1, 2, 1], [2], ZZ) == ([1], [1, 2, 1], [2])
assert dup_rr_prs_gcd([1, 2, 1], [2], ZZ) == ([1], [1, 2, 1], [2])
assert dup_zz_heu_gcd([2, 4, 2], [2], ZZ) == ([2], [1, 2, 1], [1])
assert dup_rr_prs_gcd([2, 4, 2], [2], ZZ) == ([2], [1, 2, 1], [1])
assert dup_zz_heu_gcd([2], [2, 4, 2], ZZ) == ([2], [1], [1, 2, 1])
assert dup_rr_prs_gcd([2], [2, 4, 2], ZZ) == ([2], [1], [1, 2, 1])
assert dup_zz_heu_gcd([2, 4, 2], [1, 1], ZZ) == ([1, 1], [2, 2], [1])
assert dup_rr_prs_gcd([2, 4, 2], [1, 1], ZZ) == ([1, 1], [2, 2], [1])
assert dup_zz_heu_gcd([1, 1], [2, 4, 2], ZZ) == ([1, 1], [1], [2, 2])
assert dup_rr_prs_gcd([1, 1], [2, 4, 2], ZZ) == ([1, 1], [1], [2, 2])
f, g = [1, -31], [1, 0]
assert dup_zz_heu_gcd(f, g, ZZ) == ([1], f, g)
assert dup_rr_prs_gcd(f, g, ZZ) == ([1], f, g)
f = [1, 8, 21, 22, 8]
g = [1, 6, 11, 6]
h = [1, 3, 2]
cff = [1, 5, 4]
cfg = [1, 3]
assert dup_zz_heu_gcd(f, g, ZZ) == (h, cff, cfg)
assert dup_rr_prs_gcd(f, g, ZZ) == (h, cff, cfg)
f = [1, 0, 0, 0, -4]
g = [1, 0, 4, 0, 4]
h = [1, 0, 2]
cff = [1, 0, -2]
cfg = [1, 0, 2]
assert dup_zz_heu_gcd(f, g, ZZ) == (h, cff, cfg)
assert dup_rr_prs_gcd(f, g, ZZ) == (h, cff, cfg)
f = [1, 0, 1, 0, -3, -3, 8, 2, -5]
g = [3, 0, 5, -0, -4, -9, 21]
h = [1]
cff = f
cfg = g
assert dup_zz_heu_gcd(f, g, ZZ) == (h, cff, cfg)
assert dup_rr_prs_gcd(f, g, ZZ) == (h, cff, cfg)
f = dup_normal([1, 0, 1, 0, -3, -3, 8, 2, -5], QQ)
g = dup_normal([3, 0, 5, -0, -4, -9, 21], QQ)
h = dup_normal([1], QQ)
assert dup_qq_heu_gcd(f, g, QQ) == (h, cff, cfg)
assert dup_ff_prs_gcd(f, g, QQ) == (h, cff, cfg)
f = [
-352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272,
0, 0, 0, 0, 0, 0,
46818041807522713962450042363465092040687472354933295397472942006618953623327997952,
0, 0, 0, 0, 0, 0,
378182690892293941192071663536490788434899030680411695933646320291525827756032,
0, 0, 0, 0, 0, 0,
112806468807371824947796775491032386836656074179286744191026149539708928,
0, 0, 0, 0, 0, 0,
-12278371209708240950316872681744825481125965781519138077173235712, 0,
0, 0, 0, 0, 0,
289127344604779611146960547954288113529690984687482920704, 0, 0, 0, 0,
0, 0, 19007977035740498977629742919480623972236450681, 0, 0, 0, 0, 0,
0, 311973482284542371301330321821976049
]
g = [
365431878023781158602430064717380211405897160759702125019136, 0, 0, 0,
0, 0, 0, 197599133478719444145775798221171663643171734081650688, 0, 0,
0, 0, 0, 0, -9504116979659010018253915765478924103928886144, 0, 0, 0,
0, 0, 0, -311973482284542371301330321821976049
]
f = dup_normal(f, ZZ)
g = dup_normal(g, ZZ)
assert dup_zz_heu_gcd(f, dup_diff(f, 1, ZZ), ZZ)[0] == g
assert dup_rr_prs_gcd(f, dup_diff(f, 1, ZZ), ZZ)[0] == g
f = [QQ(1, 2), QQ(1), QQ(1, 2)]
g = [QQ(1, 2), QQ(1, 2)]
h = [QQ(1), QQ(1)]
assert dup_qq_heu_gcd(f, g, QQ) == (h, g, [QQ(1, 2)])
assert dup_ff_prs_gcd(f, g, QQ) == (h, g, [QQ(1, 2)])
def test_dup_diff():
assert dup_diff([], 1, ZZ) == []
assert dup_diff([7], 1, ZZ) == []
assert dup_diff([2, 7], 1, ZZ) == [2]
assert dup_diff([1, 2, 1], 1, ZZ) == [2, 2]
assert dup_diff([1, 2, 3, 4], 1, ZZ) == [3, 4, 3]
assert dup_diff([1, -1, 0, 0, 2], 1, ZZ) == [4, -3, 0, 0]
f = dup_normal([17, 34, 56, -345, 23, 76, 0, 0, 12, 3, 7], ZZ)
assert dup_diff(f, 0, ZZ) == f
assert dup_diff(f, 1, ZZ) == [170, 306, 448, -2415, 138, 380, 0, 0, 24, 3]
assert dup_diff(f, 2, ZZ) == dup_diff(dup_diff(f, 1, ZZ), 1, ZZ)
assert dup_diff(f, 3, ZZ) == dup_diff(dup_diff(dup_diff(f, 1, ZZ), 1, ZZ),
1, ZZ)
K = FF(3)
f = dup_normal([17, 34, 56, -345, 23, 76, 0, 0, 12, 3, 7], K)
assert dup_diff(f, 1, K) == dup_normal([2, 0, 1, 0, 0, 2, 0, 0, 0, 0], K)
assert dup_diff(f, 2, K) == dup_normal([1, 0, 0, 2, 0, 0, 0], K)
assert dup_diff(f, 3, K) == dup_normal([], K)
assert dup_diff(f, 0, K) == f
assert dup_diff(f, 2, K) == dup_diff(dup_diff(f, 1, K), 1, K)
assert dup_diff(f, 3, K) == dup_diff(dup_diff(dup_diff(f, 1, K), 1, K), 1,
K)
def test_dup_diff():
assert dup_diff([], 1, ZZ) == []
assert dup_diff([7], 1, ZZ) == []
assert dup_diff([2, 7], 1, ZZ) == [2]
assert dup_diff([1, 2, 1], 1, ZZ) == [2, 2]
assert dup_diff([1, 2, 3, 4], 1, ZZ) == [3, 4, 3]
assert dup_diff([1, -1, 0, 0, 2], 1, ZZ) == [4, -3, 0, 0]
f = dup_normal([17, 34, 56, -345, 23, 76, 0, 0, 12, 3, 7], ZZ)
assert dup_diff(f, 0, ZZ) == f
assert dup_diff(f, 1, ZZ) == dup_diff(f, 1, ZZ)
assert dup_diff(f, 2, ZZ) == dup_diff(dup_diff(f, 1, ZZ), 1, ZZ)
assert dup_diff(
f, 3, ZZ) == dup_diff(dup_diff(dup_diff(f, 1, ZZ), 1, ZZ), 1, ZZ)
K = FF(3)
f = dup_normal([17, 34, 56, -345, 23, 76, 0, 0, 12, 3, 7], K)
assert dup_diff(f, 1, K) == dup_normal([2, 0, 1, 0, 0, 2, 0, 0, 0, 0], K)
assert dup_diff(f, 2, K) == dup_normal([1, 0, 0, 2, 0, 0, 0], K)
assert dup_diff(f, 3, K) == dup_normal([], K)
assert dup_diff(f, 0, K) == f
assert dup_diff(f, 1, K) == dup_diff(f, 1, K)
assert dup_diff(f, 2, K) == dup_diff(dup_diff(f, 1, K), 1, K)
assert dup_diff(
f, 3, K) == dup_diff(dup_diff(dup_diff(f, 1, K), 1, K), 1, K)
def test_dup_gcd():
assert dup_zz_heu_gcd([], [], ZZ) == ([], [], [])
assert dup_rr_prs_gcd([], [], ZZ) == ([], [], [])
assert dup_zz_heu_gcd([2], [], ZZ) == ([2], [1], [])
assert dup_rr_prs_gcd([2], [], ZZ) == ([2], [1], [])
assert dup_zz_heu_gcd([-2], [], ZZ) == ([2], [-1], [])
assert dup_rr_prs_gcd([-2], [], ZZ) == ([2], [-1], [])
assert dup_zz_heu_gcd([], [-2], ZZ) == ([2], [], [-1])
assert dup_rr_prs_gcd([], [-2], ZZ) == ([2], [], [-1])
assert dup_zz_heu_gcd([], [2,4], ZZ) == ([2,4], [], [1])
assert dup_rr_prs_gcd([], [2,4], ZZ) == ([2,4], [], [1])
assert dup_zz_heu_gcd([2,4], [], ZZ) == ([2,4], [1], [])
assert dup_rr_prs_gcd([2,4], [], ZZ) == ([2,4], [1], [])
assert dup_zz_heu_gcd([2], [2], ZZ) == ([2], [1], [1])
assert dup_rr_prs_gcd([2], [2], ZZ) == ([2], [1], [1])
assert dup_zz_heu_gcd([-2], [2], ZZ) == ([2], [-1], [1])
assert dup_rr_prs_gcd([-2], [2], ZZ) == ([2], [-1], [1])
assert dup_zz_heu_gcd([2], [-2], ZZ) == ([2], [1], [-1])
assert dup_rr_prs_gcd([2], [-2], ZZ) == ([2], [1], [-1])
assert dup_zz_heu_gcd([-2], [-2], ZZ) == ([2], [-1], [-1])
assert dup_rr_prs_gcd([-2], [-2], ZZ) == ([2], [-1], [-1])
assert dup_zz_heu_gcd([1,2,1], [1], ZZ) == ([1], [1, 2, 1], [1])
assert dup_rr_prs_gcd([1,2,1], [1], ZZ) == ([1], [1, 2, 1], [1])
assert dup_zz_heu_gcd([1,2,1], [2], ZZ) == ([1], [1, 2, 1], [2])
assert dup_rr_prs_gcd([1,2,1], [2], ZZ) == ([1], [1, 2, 1], [2])
assert dup_zz_heu_gcd([2,4,2], [2], ZZ) == ([2], [1, 2, 1], [1])
assert dup_rr_prs_gcd([2,4,2], [2], ZZ) == ([2], [1, 2, 1], [1])
assert dup_zz_heu_gcd([2], [2,4,2], ZZ) == ([2], [1], [1, 2, 1])
assert dup_rr_prs_gcd([2], [2,4,2], ZZ) == ([2], [1], [1, 2, 1])
assert dup_zz_heu_gcd([2,4,2], [1,1], ZZ) == ([1, 1], [2, 2], [1])
assert dup_rr_prs_gcd([2,4,2], [1,1], ZZ) == ([1, 1], [2, 2], [1])
assert dup_zz_heu_gcd([1,1], [2,4,2], ZZ) == ([1, 1], [1], [2, 2])
assert dup_rr_prs_gcd([1,1], [2,4,2], ZZ) == ([1, 1], [1], [2, 2])
f, g = [1, -31], [1, 0]
assert dup_zz_heu_gcd(f, g, ZZ) == ([1], f, g)
assert dup_rr_prs_gcd(f, g, ZZ) == ([1], f, g)
f = [1,8,21,22,8]
g = [1,6,11,6]
h = [1,3,2]
cff = [1,5,4]
cfg = [1,3]
assert dup_zz_heu_gcd(f, g, ZZ) == (h, cff, cfg)
assert dup_rr_prs_gcd(f, g, ZZ) == (h, cff, cfg)
f = [1,0,0,0,-4]
g = [1,0,4,0, 4]
h = [1,0,2]
cff = [1,0,-2]
cfg = [1,0, 2]
assert dup_zz_heu_gcd(f, g, ZZ) == (h, cff, cfg)
assert dup_rr_prs_gcd(f, g, ZZ) == (h, cff, cfg)
f = [1,0,1,0,-3,-3,8,2,-5]
g = [3,0,5,-0,-4,-9,21]
h = [1]
cff = f
cfg = g
assert dup_zz_heu_gcd(f, g, ZZ) == (h, cff, cfg)
assert dup_rr_prs_gcd(f, g, ZZ) == (h, cff, cfg)
f = dup_normal([1,0,1,0,-3,-3,8,2,-5], QQ)
g = dup_normal([3,0,5,-0,-4,-9,21], QQ)
h = dup_normal([1], QQ)
assert dup_qq_heu_gcd(f, g, QQ) == (h, cff, cfg)
assert dup_ff_prs_gcd(f, g, QQ) == (h, cff, cfg)
f = [-352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272,
0, 0, 0, 0, 0, 0,
46818041807522713962450042363465092040687472354933295397472942006618953623327997952,
0, 0, 0, 0, 0, 0,
378182690892293941192071663536490788434899030680411695933646320291525827756032,
0, 0, 0, 0, 0, 0,
112806468807371824947796775491032386836656074179286744191026149539708928,
0, 0, 0, 0, 0, 0,
-12278371209708240950316872681744825481125965781519138077173235712,
0, 0, 0, 0, 0, 0,
289127344604779611146960547954288113529690984687482920704,
0, 0, 0, 0, 0, 0,
19007977035740498977629742919480623972236450681,
0, 0, 0, 0, 0, 0,
311973482284542371301330321821976049]
g = [365431878023781158602430064717380211405897160759702125019136,
0, 0, 0, 0, 0, 0,
197599133478719444145775798221171663643171734081650688,
0, 0, 0, 0, 0, 0,
-9504116979659010018253915765478924103928886144,
0, 0, 0, 0, 0, 0,
-311973482284542371301330321821976049]
f = dup_normal(f, ZZ)
g = dup_normal(g, ZZ)
assert dup_zz_heu_gcd(f, dup_diff(f, 1, ZZ), ZZ)[0] == g
assert dup_rr_prs_gcd(f, dup_diff(f, 1, ZZ), ZZ)[0] == g
f = [QQ(1,2),QQ(1),QQ(1,2)]
g = [QQ(1,2),QQ(1,2)]
h = [QQ(1), QQ(1)]
assert dup_qq_heu_gcd(f, g, QQ) == (h, g, [QQ(1,2)])
assert dup_ff_prs_gcd(f, g, QQ) == (h, g, [QQ(1,2)])
