def poly_factors(f, *symbols, **flags):
"""Factor polynomials over rationals.
>>> from sympy.polys.factortools import poly_factors
>>> from sympy.abc import x, y
>>> poly_factors(x**2 - y**2, x, y)
(1, [(Poly(x - y, x, y), 1), (Poly(x + y, x, y), 1)])
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
symbols = list(f.symbols)
try:
denom, F = f.as_integer()
except CoefficientError:
other = set([])
for coeff in f.iter_coeffs():
other |= coeff.atoms(Symbol)
symbols += sorted(other)
F = Poly(f, *symbols)
denom, F = F.as_integer()
cont, factors = zzX_factor(zzX_from_poly(F))
for i, (h, k) in enumerate(factors):
h = zzX_to_poly(h, *symbols)
if f.symbols != symbols:
h = h.as_poly(*f.symbols)
factors[i] = (h, k)
return Rational(cont, denom), factors
def poly_factors(f, *symbols, **flags):
"""Factor polynomials over rationals.
>>> from sympy import *
>>> x, y = symbols("x y")
>>> poly_factors(x**2 - y**2, x, y)
(1, [(Poly(x - y, x, y), 1), (Poly(x + y, x, y), 1)])
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
symbols = list(f.symbols)
try:
denom, F = f.as_integer()
except CoefficientError:
other = set([])
for coeff in f.iter_coeffs():
other |= coeff.atoms(Symbol)
symbols += sorted(other)
F = Poly(f, *symbols)
denom, F = F.as_integer()
cont, factors = zzX_factor(zzX_from_poly(F))
for i, (h, k) in enumerate(factors):
h = zzX_to_poly(h, *symbols)
if f.symbols != symbols:
h = h.as_poly(*f.symbols)
factors[i] = (h, k)
return Rational(cont, denom), factors
def test_zzX_from_to():
assert zzX_from_dict({}, 2) == [[]]
f = [3,0,0,2,0,0,0,0,8]
g = (3*x**8 + 2*x**5 + 8).as_poly(x)
assert zzX_from_poly(g) == f
assert zzX_to_poly(f, x) == g
assert zzX_from_poly(F_0) == f_0
assert zzX_from_poly(F_1) == f_1
assert zzX_from_poly(F_2) == f_2
assert zzX_from_poly(F_3) == f_3
assert zzX_from_poly(F_4) == f_4
assert zzX_from_poly(F_5) == f_5
assert zzX_from_poly(F_6) == f_6
assert zzX_to_poly(f_0, x,y,z) == F_0
assert zzX_to_poly(f_1, x,y,z) == F_1
assert zzX_to_poly(f_2, x,y,z) == F_2
assert zzX_to_poly(f_3, x,y,z) == F_3
assert zzX_to_poly(f_4, x,y,z) == F_4
assert zzX_to_poly(f_5, x,y,z) == F_5
assert zzX_to_poly(f_6, x,y,z,t) == F_6
def test_zzX_from_to():
assert zzX_from_dict({}, 2) == [[]]
f = [3, 0, 0, 2, 0, 0, 0, 0, 8]
g = (3 * x**8 + 2 * x**5 + 8).as_poly(x)
assert zzX_from_poly(g) == f
assert zzX_to_poly(f, x) == g
assert zzX_from_poly(F_0) == f_0
assert zzX_from_poly(F_1) == f_1
assert zzX_from_poly(F_2) == f_2
assert zzX_from_poly(F_3) == f_3
assert zzX_from_poly(F_4) == f_4
assert zzX_from_poly(F_5) == f_5
assert zzX_from_poly(F_6) == f_6
assert zzX_to_poly(f_0, x, y, z) == F_0
assert zzX_to_poly(f_1, x, y, z) == F_1
assert zzX_to_poly(f_2, x, y, z) == F_2
assert zzX_to_poly(f_3, x, y, z) == F_3
assert zzX_to_poly(f_4, x, y, z) == F_4
assert zzX_to_poly(f_5, x, y, z) == F_5
assert zzX_to_poly(f_6, x, y, z, t) == F_6
def test_zzX_wang():
f = zzX_from_poly(W_1)
p = nextprime(zzX_mignotte_bound(f))
assert p == 6291469
V_1, k_1, E_1 = [[1],[]], 1, -14
V_2, k_2, E_2 = [[1, 0]], 2, 3
V_3, k_3, E_3 = [[1],[ 1, 0]], 2, -11
V_4, k_4, E_4 = [[1],[-1, 0]], 1, -17
V = [V_1, V_2, V_3, V_4]
K = [k_1, k_2, k_3, k_4]
E = [E_1, E_2, E_3, E_4]
V = zip(V, K)
A = [-14, 3]
U = zzX_eval_list(f, A)
cu, u = zzx_primitive(U)
assert cu == 1 and u == U == \
[1036728, 915552, 55748, 105621, -17304, -26841, -644]
assert zzX_wang_non_divisors(E, cu, 4) == [7, 3, 11, 17]
assert zzx_sqf_p(u) and zzx_degree(u) == zzX_degree(f)
_, H = zzx_factor_sqf(u)
h_1 = [44, 42, 1]
h_2 = [126, -9, 28]
h_3 = [187, 0, -23]
assert H == [h_1, h_2, h_3]
LC_1 = [[-4], [-4,0]]
LC_2 = [[-1,0,0], []]
LC_3 = [[1], [], [-1,0,0]]
LC = [LC_1, LC_2, LC_3]
assert zzX_wang_lead_coeffs(f, V, cu, E, H, A) == (f, H, LC)
H_1 = [[44L, 42L, 1L], [126L, -9L, 28L], [187L, 0L, -23L]]
C_1 = [-70686, -5863, -17826, 2009, 5031, 74]
H_2 = [[[-4, -12], [-3, 0], [1]], [[-9, 0], [-9], [-2, 0]], [[1, 0, -9], [], [1, -9]]]
C_2 = [[9, 12, -45, -108, -324], [18, -216, -810, 0], [2, 9, -252, -288, -945], [-30, -414, 0], [2, -54, -3, 81], [12, 0]]
H_3 = [[[-4, -12], [-3, 0], [1]], [[-9, 0], [-9], [-2, 0]], [[1, 0, -9], [], [1, -9]]]
C_3 = [[-36, -108, 0], [-27, -36, -108], [-8, -42, 0], [-6, 0, 9], [2, 0]]
T_1 = [[-3, 0], [-2], [1]]
T_2 = [[[-1, 0], []], [[-3], []], [[-6]]]
T_3 = [[[]], [[]], [[-1]]]
assert zzX_diophantine(H_1, C_1, [], 5, p) == T_1
assert zzX_diophantine(H_2, C_2, [-14], 5, p) == T_2
assert zzX_diophantine(H_3, C_3, [-14], 5, p) == T_3
factors = zzX_wang_hensel_lifting(f, H, LC, A, p)
f_1 = zzX_to_poly(factors[0], x, y, z)
f_2 = zzX_to_poly(factors[1], x, y, z)
f_3 = zzX_to_poly(factors[2], x, y, z)
assert f_1 == -(4*(y + z)*x**2 + x*y*z - 1).as_poly(x, y, z)
assert f_2 == -(y*z**2*x**2 + 3*x*z + 2*y).as_poly(x, y, z)
assert f_3 == ((y**2 - z**2)*x**2 + y - z**2).as_poly(x, y, z)
assert f_1*f_2*f_3 == W_1
def test_zzX_wang():
f = zzX_from_poly(W_1)
p = nextprime(zzX_mignotte_bound(f))
assert p == 6291469
V_1, k_1, E_1 = [[1], []], 1, -14
V_2, k_2, E_2 = [[1, 0]], 2, 3
V_3, k_3, E_3 = [[1], [1, 0]], 2, -11
V_4, k_4, E_4 = [[1], [-1, 0]], 1, -17
V = [V_1, V_2, V_3, V_4]
K = [k_1, k_2, k_3, k_4]
E = [E_1, E_2, E_3, E_4]
V = zip(V, K)
A = [-14, 3]
U = zzX_eval_list(f, A)
cu, u = zzx_primitive(U)
assert cu == 1 and u == U == \
[1036728, 915552, 55748, 105621, -17304, -26841, -644]
assert zzX_wang_non_divisors(E, cu, 4) == [7, 3, 11, 17]
assert zzx_sqf_p(u) and zzx_degree(u) == zzX_degree(f)
_, H = zzx_factor_sqf(u)
h_1 = [44, 42, 1]
h_2 = [126, -9, 28]
h_3 = [187, 0, -23]
assert H == [h_1, h_2, h_3]
LC_1 = [[-4], [-4, 0]]
LC_2 = [[-1, 0, 0], []]
LC_3 = [[1], [], [-1, 0, 0]]
LC = [LC_1, LC_2, LC_3]
assert zzX_wang_lead_coeffs(f, V, cu, E, H, A) == (f, H, LC)
H_1 = [[44L, 42L, 1L], [126L, -9L, 28L], [187L, 0L, -23L]]
C_1 = [-70686, -5863, -17826, 2009, 5031, 74]
H_2 = [[[-4, -12], [-3, 0], [1]], [[-9, 0], [-9], [-2, 0]],
[[1, 0, -9], [], [1, -9]]]
C_2 = [[9, 12, -45, -108, -324], [18, -216, -810, 0],
[2, 9, -252, -288, -945], [-30, -414, 0], [2, -54, -3, 81], [12, 0]]
H_3 = [[[-4, -12], [-3, 0], [1]], [[-9, 0], [-9], [-2, 0]],
[[1, 0, -9], [], [1, -9]]]
C_3 = [[-36, -108, 0], [-27, -36, -108], [-8, -42, 0], [-6, 0, 9], [2, 0]]
T_1 = [[-3, 0], [-2], [1]]
T_2 = [[[-1, 0], []], [[-3], []], [[-6]]]
T_3 = [[[]], [[]], [[-1]]]
assert zzX_diophantine(H_1, C_1, [], 5, p) == T_1
assert zzX_diophantine(H_2, C_2, [-14], 5, p) == T_2
assert zzX_diophantine(H_3, C_3, [-14], 5, p) == T_3
factors = zzX_wang_hensel_lifting(f, H, LC, A, p)
f_1 = zzX_to_poly(factors[0], x, y, z)
f_2 = zzX_to_poly(factors[1], x, y, z)
f_3 = zzX_to_poly(factors[2], x, y, z)
assert f_1 == -(4 * (y + z) * x**2 + x * y * z - 1).as_poly(x, y, z)
assert f_2 == -(y * z**2 * x**2 + 3 * x * z + 2 * y).as_poly(x, y, z)
assert f_3 == ((y**2 - z**2) * x**2 + y - z**2).as_poly(x, y, z)
assert f_1 * f_2 * f_3 == W_1
