def dmp_zz_wang_hensel_lifting(f, H, LC, A, p, u, K):
"""Wang/EEZ: Parallel Hensel lifting algorithm. """
S, n, v = [f], len(A), u-1
H = list(H)
for i, a in enumerate(reversed(A[1:])):
s = dmp_eval_in(S[0], a, n-i, u-i, K)
S.insert(0, dmp_ground_trunc(s, p, v-i, K))
d = max(dmp_degree_list(f, u)[1:])
for j, s, a in zip(xrange(2, n+2), S, A):
G, w = list(H), j-1
I, J = A[:j-2], A[j-1:]
for i, (h, lc) in enumerate(zip(H, LC)):
lc = dmp_ground_trunc(dmp_eval_tail(lc, J, v, K), p, w-1, K)
H[i] = [lc] + dmp_raise(h[1:], 1, w-1, K)
m = dmp_nest([K.one, -a], w, K)
M = dmp_one(w, K)
c = dmp_sub(s, dmp_expand(H, w, K), w, K)
dj = dmp_degree_in(s, w, w)
for k in xrange(0, dj):
if dmp_zero_p(c, w):
break
M = dmp_mul(M, m, w, K)
C = dmp_diff_eval_in(c, k+1, a, w, w, K)
if not dmp_zero_p(C, w-1):
C = dmp_quo_ground(C, K.factorial(k+1), w-1, K)
T = dmp_zz_diophantine(G, C, I, d, p, w-1, K)
for i, (h, t) in enumerate(zip(H, T)):
h = dmp_add_mul(h, dmp_raise(t, 1, w-1, K), M, w, K)
H[i] = dmp_ground_trunc(h, p, w, K)
h = dmp_sub(s, dmp_expand(H, w, K), w, K)
c = dmp_ground_trunc(h, p, w, K)
if dmp_expand(H, u, K) != f:
raise ExtraneousFactors # pragma: no cover
else:
return H
def dmp_zz_wang_hensel_lifting(f, H, LC, A, p, u, K):
"""Wang/EEZ: Parallel Hensel lifting algorithm. """
S, n, v = [f], len(A), u - 1
H = list(H)
for i, a in enumerate(reversed(A[1:])):
s = dmp_eval_in(S[0], a, n - i, u - i, K)
S.insert(0, dmp_ground_trunc(s, p, v - i, K))
d = max(dmp_degree_list(f, u)[1:])
for j, s, a in zip(xrange(2, n + 2), S, A):
G, w = list(H), j - 1
I, J = A[:j - 2], A[j - 1:]
for i, (h, lc) in enumerate(zip(H, LC)):
lc = dmp_ground_trunc(dmp_eval_tail(lc, J, v, K), p, w - 1, K)
H[i] = [lc] + dmp_raise(h[1:], 1, w - 1, K)
m = dmp_nest([K.one, -a], w, K)
M = dmp_one(w, K)
c = dmp_sub(s, dmp_expand(H, w, K), w, K)
dj = dmp_degree_in(s, w, w)
for k in xrange(0, dj):
if dmp_zero_p(c, w):
break
M = dmp_mul(M, m, w, K)
C = dmp_diff_eval_in(c, k + 1, a, w, w, K)
if not dmp_zero_p(C, w - 1):
C = dmp_quo_ground(C, K.factorial(k + 1), w - 1, K)
T = dmp_zz_diophantine(G, C, I, d, p, w - 1, K)
for i, (h, t) in enumerate(zip(H, T)):
h = dmp_add_mul(h, dmp_raise(t, 1, w - 1, K), M, w, K)
H[i] = dmp_ground_trunc(h, p, w, K)
h = dmp_sub(s, dmp_expand(H, w, K), w, K)
c = dmp_ground_trunc(h, p, w, K)
if dmp_expand(H, u, K) != f:
raise ExtraneousFactors # pragma: no cover
else:
return H
def dmp_sqf_list(f, u, 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 dmp_sqf_list
>>> f = ZZ.map([[1], [2, 0], [1, 0, 0], [], [], []])
>>> dmp_sqf_list(f, 1, ZZ)
(1, [([[1], [1, 0]], 2), ([[1], []], 3)])
>>> dmp_sqf_list(f, 1, ZZ, all=True)
(1, [([[1]], 1), ([[1], [1, 0]], 2), ([[1], []], 3)])
"""
if not u:
return dup_sqf_list(f, K, all=all)
if not K.has_CharacteristicZero:
return dmp_gf_sqf_list(f, u, K, all=all)
if K.has_Field or not K.is_Exact:
coeff = dmp_ground_LC(f, u, K)
f = dmp_ground_monic(f, u, K)
else:
coeff, f = dmp_ground_primitive(f, u, K)
if K.is_negative(dmp_ground_LC(f, u, K)):
f = dmp_neg(f, u, K)
coeff = -coeff
if dmp_degree(f, u) <= 0:
return coeff, []
result, i = [], 1
h = dmp_diff(f, 1, u, K)
g, p, q = dmp_inner_gcd(f, h, u, K)
while True:
d = dmp_diff(p, 1, u, K)
h = dmp_sub(q, d, u, K)
if dmp_zero_p(h, u):
result.append((p, i))
break
g, p, q = dmp_inner_gcd(p, h, u, K)
if all or dmp_degree(g, u) > 0:
result.append((g, i))
i += 1
return coeff, result
def dmp_sqf_list(f, u, 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 dmp_sqf_list
>>> f = ZZ.map([[1], [2, 0], [1, 0, 0], [], [], []])
>>> dmp_sqf_list(f, 1, ZZ)
(1, [([[1], [1, 0]], 2), ([[1], []], 3)])
>>> dmp_sqf_list(f, 1, ZZ, all=True)
(1, [([[1]], 1), ([[1], [1, 0]], 2), ([[1], []], 3)])
"""
if not u:
return dup_sqf_list(f, K, all=all)
if not K.has_CharacteristicZero:
return dmp_gf_sqf_list(f, u, K, all=all)
if K.has_Field or not K.is_Exact:
coeff = dmp_ground_LC(f, u, K)
f = dmp_ground_monic(f, u, K)
else:
coeff, f = dmp_ground_primitive(f, u, K)
if K.is_negative(dmp_ground_LC(f, u, K)):
f = dmp_neg(f, u, K)
coeff = -coeff
if dmp_degree(f, u) <= 0:
return coeff, []
result, i = [], 1
h = dmp_diff(f, 1, u, K)
g, p, q = dmp_inner_gcd(f, h, u, K)
while True:
d = dmp_diff(p, 1, u, K)
h = dmp_sub(q, d, u, K)
if dmp_zero_p(h, u):
result.append((p, i))
break
g, p, q = dmp_inner_gcd(p, h, u, K)
if all or dmp_degree(g, u) > 0:
result.append((g, i))
i += 1
return coeff, result
def dup_real_imag(f, K):
"""
Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dup_real_imag
>>> dup_real_imag([ZZ(1), ZZ(1), ZZ(1), ZZ(1)], ZZ)
([[1], [1], [-3, 0, 1], [-1, 0, 1]], [[3, 0], [2, 0], [-1, 0, 1, 0]])
"""
if not K.is_ZZ and not K.is_QQ:
raise DomainError(
"computing real and imaginary parts is not supported over %s" % K)
f1 = dmp_zero(1)
f2 = dmp_zero(1)
if not f:
return f1, f2
g = [[[K.one, K.zero]], [[K.one], []]]
h = dmp_ground(f[0], 2)
for c in f[1:]:
h = dmp_mul(h, g, 2, K)
h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)
H = dup_to_raw_dict(h)
for k, h in H.iteritems():
m = k % 4
if not m:
f1 = dmp_add(f1, h, 1, K)
elif m == 1:
f2 = dmp_add(f2, h, 1, K)
elif m == 2:
f1 = dmp_sub(f1, h, 1, K)
else:
f2 = dmp_sub(f2, h, 1, K)
return f1, f2
def dup_real_imag(f, K):
"""
Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dup_real_imag
>>> dup_real_imag([ZZ(1), ZZ(1), ZZ(1), ZZ(1)], ZZ)
([[1], [1], [-3, 0, 1], [-1, 0, 1]], [[3, 0], [2, 0], [-1, 0, 1, 0]])
"""
if not K.is_ZZ and not K.is_QQ:
raise DomainError(
"computing real and imaginary parts is not supported over %s" % K)
f1 = dmp_zero(1)
f2 = dmp_zero(1)
if not f:
return f1, f2
g = [[[K.one, K.zero]], [[K.one], []]]
h = dmp_ground(f[0], 2)
for c in f[1:]:
h = dmp_mul(h, g, 2, K)
h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)
H = dup_to_raw_dict(h)
for k, h in H.iteritems():
m = k % 4
if not m:
f1 = dmp_add(f1, h, 1, K)
elif m == 1:
f2 = dmp_add(f2, h, 1, K)
elif m == 2:
f1 = dmp_sub(f1, h, 1, K)
else:
f2 = dmp_sub(f2, h, 1, K)
return f1, f2
def dup_real_imag(f, K):
"""
Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> R.dup_real_imag(x**3 + x**2 + x + 1)
(x**3 + x**2 - 3*x*y**2 + x - y**2 + 1, 3*x**2*y + 2*x*y - y**3 + y)
"""
if not K.is_ZZ and not K.is_QQ:
raise DomainError(
"computing real and imaginary parts is not supported over %s" % K)
f1 = dmp_zero(1)
f2 = dmp_zero(1)
if not f:
return f1, f2
g = [[[K.one, K.zero]], [[K.one], []]]
h = dmp_ground(f[0], 2)
for c in f[1:]:
h = dmp_mul(h, g, 2, K)
h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)
H = dup_to_raw_dict(h)
for k, h in H.items():
m = k % 4
if not m:
f1 = dmp_add(f1, h, 1, K)
elif m == 1:
f2 = dmp_add(f2, h, 1, K)
elif m == 2:
f1 = dmp_sub(f1, h, 1, K)
else:
f2 = dmp_sub(f2, h, 1, K)
return f1, f2
def dmp_sqf_list(f, u, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[X]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x**5 + 2*x**4*y + x**3*y**2
>>> R.dmp_sqf_list(f)
(1, [(x + y, 2), (x, 3)])
>>> R.dmp_sqf_list(f, all=True)
(1, [(1, 1), (x + y, 2), (x, 3)])
"""
if not u:
return dup_sqf_list(f, K, all=all)
if K.is_FiniteField:
return dmp_gf_sqf_list(f, u, K, all=all)
if K.is_Field:
coeff = dmp_ground_LC(f, u, K)
f = dmp_ground_monic(f, u, K)
else:
coeff, f = dmp_ground_primitive(f, u, K)
if K.is_negative(dmp_ground_LC(f, u, K)):
f = dmp_neg(f, u, K)
coeff = -coeff
if dmp_degree(f, u) <= 0:
return coeff, []
result, i = [], 1
h = dmp_diff(f, 1, u, K)
g, p, q = dmp_inner_gcd(f, h, u, K)
while True:
d = dmp_diff(p, 1, u, K)
h = dmp_sub(q, d, u, K)
if dmp_zero_p(h, u):
result.append((p, i))
break
g, p, q = dmp_inner_gcd(p, h, u, K)
if all or dmp_degree(g, u) > 0:
result.append((g, i))
i += 1
return coeff, result
def dmp_sqf_list(f, u, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[X]``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x**5 + 2*x**4*y + x**3*y**2
>>> R.dmp_sqf_list(f)
(1, [(x + y, 2), (x, 3)])
>>> R.dmp_sqf_list(f, all=True)
(1, [(1, 1), (x + y, 2), (x, 3)])
"""
if not u:
return dup_sqf_list(f, K, all=all)
if K.is_FiniteField:
return dmp_gf_sqf_list(f, u, K, all=all)
if K.has_Field:
coeff = dmp_ground_LC(f, u, K)
f = dmp_ground_monic(f, u, K)
else:
coeff, f = dmp_ground_primitive(f, u, K)
if K.is_negative(dmp_ground_LC(f, u, K)):
f = dmp_neg(f, u, K)
coeff = -coeff
if dmp_degree(f, u) <= 0:
return coeff, []
result, i = [], 1
h = dmp_diff(f, 1, u, K)
g, p, q = dmp_inner_gcd(f, h, u, K)
while True:
d = dmp_diff(p, 1, u, K)
h = dmp_sub(q, d, u, K)
if dmp_zero_p(h, u):
result.append((p, i))
break
g, p, q = dmp_inner_gcd(p, h, u, K)
if all or dmp_degree(g, u) > 0:
result.append((g, i))
i += 1
return coeff, result
def dup_real_imag(f, K):
"""
Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> R.dup_real_imag(x**3 + x**2 + x + 1)
(x**3 + x**2 - 3*x*y**2 + x - y**2 + 1, 3*x**2*y + 2*x*y - y**3 + y)
"""
if not K.is_ZZ and not K.is_QQ:
raise DomainError("computing real and imaginary parts is not supported over %s" % K)
f1 = dmp_zero(1)
f2 = dmp_zero(1)
if not f:
return f1, f2
g = [[[K.one, K.zero]], [[K.one], []]]
h = dmp_ground(f[0], 2)
for c in f[1:]:
h = dmp_mul(h, g, 2, K)
h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)
H = dup_to_raw_dict(h)
for k, h in H.items():
m = k % 4
if not m:
f1 = dmp_add(f1, h, 1, K)
elif m == 1:
f2 = dmp_add(f2, h, 1, K)
elif m == 2:
f1 = dmp_sub(f1, h, 1, K)
else:
f2 = dmp_sub(f2, h, 1, K)
return f1, f2
def sub(f, g):
"""Subtract two multivariate fractions `f` and `g`. """
if isinstance(g, DMP):
lev, dom, per, (F_num, F_den), G = f.poly_unify(g)
num, den = dmp_sub_mul(F_num, F_den, G, lev, dom), F_den
else:
lev, dom, per, F, G = f.frac_unify(g)
(F_num, F_den), (G_num, G_den) = F, G
num = dmp_sub(dmp_mul(F_num, G_den, lev, dom), dmp_mul(F_den, G_num, lev, dom), lev, dom)
den = dmp_mul(F_den, G_den, lev, dom)
return per(num, den)
def sub(f, g):
"""Subtract two multivariate fractions `f` and `g`. """
if isinstance(g, DMP):
lev, dom, per, (F_num, F_den), G = f.poly_unify(g)
num, den = dmp_sub_mul(F_num, F_den, G, lev, dom), F_den
else:
lev, dom, per, F, G = f.frac_unify(g)
(F_num, F_den), (G_num, G_den) = F, G
num = dmp_sub(dmp_mul(F_num, G_den, lev, dom),
dmp_mul(F_den, G_num, lev, dom), lev, dom)
den = dmp_mul(F_den, G_den, lev, dom)
return per(num, den)
def _dmp_zz_gcd_interpolate(h, x, v, K):
"""Interpolate polynomial GCD from integer GCD. """
f = []
while not dmp_zero_p(h, v):
g = dmp_ground_trunc(h, x, v, K)
f.insert(0, g)
h = dmp_sub(h, g, v, K)
h = dmp_exquo_ground(h, x, v, K)
if K.is_negative(dmp_ground_LC(f, v+1, K)):
return dmp_neg(f, v+1, K)
else:
return f
def _dmp_zz_gcd_interpolate(h, x, v, K):
"""Interpolate polynomial GCD from integer GCD. """
f = []
while not dmp_zero_p(h, v):
g = dmp_ground_trunc(h, x, v, K)
f.insert(0, g)
h = dmp_sub(h, g, v, K)
h = dmp_quo_ground(h, x, v, K)
if K.is_negative(dmp_ground_LC(f, v + 1, K)):
return dmp_neg(f, v + 1, K)
else:
return f
def dmp_zz_modular_resultant(f, g, p, u, K):
"""
Compute resultant of `f` and `g` modulo a prime `p`.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dmp_zz_modular_resultant
>>> f = ZZ.map([[1], [1, 2]])
>>> g = ZZ.map([[2, 1], [3]])
>>> dmp_zz_modular_resultant(f, g, ZZ(5), 1, ZZ)
[-2, 0, 1]
"""
if not u:
return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)
v = u - 1
n = dmp_degree(f, u)
m = dmp_degree(g, u)
N = dmp_degree_in(f, 1, u)
M = dmp_degree_in(g, 1, u)
B = n * M + m * N
D, a = [K.one], -K.one
r = dmp_zero(v)
while dup_degree(D) <= B:
while True:
a += K.one
if a == p:
raise HomomorphismFailed('no luck')
F = dmp_eval_in(f, gf_int(a, p), 1, u, K)
if dmp_degree(F, v) == n:
G = dmp_eval_in(g, gf_int(a, p), 1, u, K)
if dmp_degree(G, v) == m:
break
R = dmp_zz_modular_resultant(F, G, p, v, K)
e = dmp_eval(r, a, v, K)
if not v:
R = dup_strip([R])
e = dup_strip([e])
else:
R = [R]
e = [e]
d = K.invert(dup_eval(D, a, K), p)
d = dup_mul_ground(D, d, K)
d = dmp_raise(d, v, 0, K)
c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
r = dmp_add(r, c, v, K)
r = dmp_ground_trunc(r, p, v, K)
D = dup_mul(D, [K.one, -a], K)
D = dup_trunc(D, p, K)
return r
def dmp_zz_modular_resultant(f, g, p, u, K):
"""
Compute resultant of ``f`` and ``g`` modulo a prime ``p``.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.euclidtools import dmp_zz_modular_resultant
>>> f = ZZ.map([[1], [1, 2]])
>>> g = ZZ.map([[2, 1], [3]])
>>> dmp_zz_modular_resultant(f, g, ZZ(5), 1, ZZ)
[-2, 0, 1]
"""
if not u:
return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)
v = u - 1
n = dmp_degree(f, u)
m = dmp_degree(g, u)
N = dmp_degree_in(f, 1, u)
M = dmp_degree_in(g, 1, u)
B = n*M + m*N
D, a = [K.one], -K.one
r = dmp_zero(v)
while dup_degree(D) <= B:
while True:
a += K.one
if a == p:
raise HomomorphismFailed('no luck')
F = dmp_eval_in(f, gf_int(a, p), 1, u, K)
if dmp_degree(F, v) == n:
G = dmp_eval_in(g, gf_int(a, p), 1, u, K)
if dmp_degree(G, v) == m:
break
R = dmp_zz_modular_resultant(F, G, p, v, K)
e = dmp_eval(r, a, v, K)
if not v:
R = dup_strip([R])
e = dup_strip([e])
else:
R = [R]
e = [e]
d = K.invert(dup_eval(D, a, K), p)
d = dup_mul_ground(D, d, K)
d = dmp_raise(d, v, 0, K)
c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
r = dmp_add(r, c, v, K)
r = dmp_ground_trunc(r, p, v, K)
D = dup_mul(D, [K.one, -a], K)
D = dup_trunc(D, p, K)
return r
def sub(f, g):
"""Subtract two multivariate polynomials `f` and `g`. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_sub(F, G, lev, dom))
def sub(f, g):
"""Subtract two multivariate polynomials `f` and `g`. """
lev, dom, per, F, G = f.unify(g)
return per(dmp_sub(F, G, lev, dom))
def test_dmp_sub():
assert dmp_sub([ZZ(1),ZZ(2)], [ZZ(1)], 0, ZZ) == \
dup_sub([ZZ(1),ZZ(2)], [ZZ(1)], ZZ)
assert dmp_sub([QQ(1,2),QQ(2,3)], [QQ(1)], 0, QQ) == \
dup_sub([QQ(1,2),QQ(2,3)], [QQ(1)], QQ)
assert dmp_sub([[[]]], [[[]]], 2, ZZ) == [[[]]]
assert dmp_sub([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_sub([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(-1)]]]
assert dmp_sub([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_sub([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(-1)]]]
assert dmp_sub([[[]]], [[[]]], 2, QQ) == [[[]]]
assert dmp_sub([[[QQ(1,2)]]], [[[]]], 2, QQ) == [[[QQ(1,2)]]]
assert dmp_sub([[[]]], [[[QQ(1,2)]]], 2, QQ) == [[[QQ(-1,2)]]]
assert dmp_sub([[[QQ(2,7)]]], [[[QQ(1,7)]]], 2, QQ) == [[[QQ(1,7)]]]
assert dmp_sub([[[QQ(1,7)]]], [[[QQ(2,7)]]], 2, QQ) == [[[QQ(-1,7)]]]
def test_dup_sqf():
assert dup_sqf_part([], ZZ) == []
assert dup_sqf_p([], ZZ) == True
assert dup_sqf_part([7], ZZ) == [1]
assert dup_sqf_p([7], ZZ) == True
assert dup_sqf_part([2,2], ZZ) == [1,1]
assert dup_sqf_p([2,2], ZZ) == True
assert dup_sqf_part([1,0,1,1], ZZ) == [1,0,1,1]
assert dup_sqf_p([1,0,1,1], ZZ) == True
assert dup_sqf_part([-1,0,1,1], ZZ) == [1,0,-1,-1]
assert dup_sqf_p([-1,0,1,1], ZZ) == True
assert dup_sqf_part([2,3,0,0], ZZ) == [2,3,0]
assert dup_sqf_p([2,3,0,0], ZZ) == False
assert dup_sqf_part([-2,3,0,0], ZZ) == [2,-3,0]
assert dup_sqf_p([-2,3,0,0], ZZ) == False
assert dup_sqf_list([], ZZ) == (0, [])
assert dup_sqf_list([1], ZZ) == (1, [])
assert dup_sqf_list([1,0], ZZ) == (1, [([1,0], 1)])
assert dup_sqf_list([2,0,0], ZZ) == (2, [([1,0], 2)])
assert dup_sqf_list([3,0,0,0], ZZ) == (3, [([1,0], 3)])
assert dup_sqf_list([ZZ(2),ZZ(4),ZZ(2)], ZZ) == \
(ZZ(2), [([ZZ(1),ZZ(1)], 2)])
assert dup_sqf_list([QQ(2),QQ(4),QQ(2)], QQ) == \
(QQ(2), [([QQ(1),QQ(1)], 2)])
assert dup_sqf_list([-1,1,0,0,1,-1], ZZ) == \
(-1, [([1,1,1,1], 1), ([1,-1], 2)])
assert dup_sqf_list([1,0,6,0,12,0,8,0,0], ZZ) == \
(1, [([1,0], 2), ([1,0,2], 3)])
K = FF(2)
f = map(K, [1,0,1])
assert dup_sqf_list(f, K) == \
(K(1), [([K(1),K(1)], 2)])
K = FF(3)
f = map(K, [1,0,0,2,0,0,2,0,0,1,0])
assert dup_sqf_list(f, K) == \
(K(1), [([K(1), K(0)], 1),
([K(1), K(1)], 3),
([K(1), K(2)], 6)])
f = [1,0,0,1]
g = map(K, f)
assert dup_sqf_part(f, ZZ) == f
assert dup_sqf_part(g, K) == [K(1), K(1)]
assert dup_sqf_p(f, ZZ) == True
assert dup_sqf_p(g, K) == False
A = [[1],[],[-3],[],[6]]
D = [[1],[],[-5],[],[5],[],[4]]
f, g = D, dmp_sub(A, dmp_mul(dmp_diff(D, 1, 1, ZZ), [[1,0]], 1, ZZ), 1, ZZ)
res = dmp_resultant(f, g, 1, ZZ)
assert dup_sqf_list(res, ZZ) == (45796, [([4,0,1], 3)])
assert dup_sqf_list_include([DMP([1, 0, 0, 0], ZZ), DMP([], ZZ), DMP([], ZZ)], ZZ[x]) == \
[([DMP([1, 0, 0, 0], ZZ)], 1), ([DMP([1], ZZ), DMP([], ZZ)], 2)]
def dmp_zz_modular_resultant(f, g, p, u, K):
"""
Compute resultant of `f` and `g` modulo a prime `p`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x + y + 2
>>> g = 2*x*y + x + 3
>>> R.dmp_zz_modular_resultant(f, g, 5)
-2*y**2 + 1
"""
if not u:
return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)
v = u - 1
n = dmp_degree(f, u)
m = dmp_degree(g, u)
N = dmp_degree_in(f, 1, u)
M = dmp_degree_in(g, 1, u)
B = n*M + m*N
D, a = [K.one], -K.one
r = dmp_zero(v)
while dup_degree(D) <= B:
while True:
a += K.one
if a == p:
raise HomomorphismFailed('no luck')
F = dmp_eval_in(f, gf_int(a, p), 1, u, K)
if dmp_degree(F, v) == n:
G = dmp_eval_in(g, gf_int(a, p), 1, u, K)
if dmp_degree(G, v) == m:
break
R = dmp_zz_modular_resultant(F, G, p, v, K)
e = dmp_eval(r, a, v, K)
if not v:
R = dup_strip([R])
e = dup_strip([e])
else:
R = [R]
e = [e]
d = K.invert(dup_eval(D, a, K), p)
d = dup_mul_ground(D, d, K)
d = dmp_raise(d, v, 0, K)
c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
r = dmp_add(r, c, v, K)
r = dmp_ground_trunc(r, p, v, K)
D = dup_mul(D, [K.one, -a], K)
D = dup_trunc(D, p, K)
return r
def dmp_zz_modular_resultant(f, g, p, u, K):
"""
Compute resultant of `f` and `g` modulo a prime `p`.
Examples
========
>>> from sympy.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = x + y + 2
>>> g = 2*x*y + x + 3
>>> R.dmp_zz_modular_resultant(f, g, 5)
-2*y**2 + 1
"""
if not u:
return gf_int(dup_prs_resultant(f, g, K)[0] % p, p)
v = u - 1
n = dmp_degree(f, u)
m = dmp_degree(g, u)
N = dmp_degree_in(f, 1, u)
M = dmp_degree_in(g, 1, u)
B = n*M + m*N
D, a = [K.one], -K.one
r = dmp_zero(v)
while dup_degree(D) <= B:
while True:
a += K.one
if a == p:
raise HomomorphismFailed('no luck')
F = dmp_eval_in(f, gf_int(a, p), 1, u, K)
if dmp_degree(F, v) == n:
G = dmp_eval_in(g, gf_int(a, p), 1, u, K)
if dmp_degree(G, v) == m:
break
R = dmp_zz_modular_resultant(F, G, p, v, K)
e = dmp_eval(r, a, v, K)
if not v:
R = dup_strip([R])
e = dup_strip([e])
else:
R = [R]
e = [e]
d = K.invert(dup_eval(D, a, K), p)
d = dup_mul_ground(D, d, K)
d = dmp_raise(d, v, 0, K)
c = dmp_mul(d, dmp_sub(R, e, v, K), v, K)
r = dmp_add(r, c, v, K)
r = dmp_ground_trunc(r, p, v, K)
D = dup_mul(D, [K.one, -a], K)
D = dup_trunc(D, p, K)
return r
