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")
denom, f = f.as_integer()
if f.is_univariate:
coeffs = map(int, f.iter_all_coeffs())
content, factors = zzx_factor(coeffs)
for i in xrange(len(factors)):
factor, k = factors[i]
n = zzx_degree(factor)
terms = {}
for j, coeff in enumerate(factor):
if coeff != 0:
terms[(n - j, )] = Integer(coeff)
factors[i] = Poly(terms, *f.symbols), k
else:
content, factors = kronecker_mv(f, **flags)
return Rational(content, 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")
denom, f = f.as_integer()
if f.is_univariate:
coeffs = map(int, f.iter_all_coeffs())
content, factors = zzx_factor(coeffs)
for i in xrange(len(factors)):
factor, k = factors[i]
n = zzx_degree(factor)
terms = {}
for j, coeff in enumerate(factor):
if coeff != 0:
terms[(n-j,)] = Integer(coeff)
factors[i] = Poly(terms, *f.symbols), k
else:
content, factors = kronecker_mv(f, **flags)
return Rational(content, denom), factors
def test_zzx_degree():
assert zzx_degree([]) == -1
assert zzx_degree([1]) == 0
assert zzx_degree([1,0]) == 1
assert zzx_degree([1,0,0,0,1]) == 4
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 kronecker_mv(f, **flags):
"""Kronecker method for Z[X] polynomials.
NOTE: This function is very slow even on small input.
Use debug=True flag to see its progress, if any.
"""
symbols = f.symbols
def mv_int_div(f, g):
q = Poly((), *symbols)
r = Poly((), *symbols)
while not f.is_zero:
lc_f, lc_g = f.LC, g.LC
dv = lc_f % lc_g
cf = lc_f / lc_g
monom = monomial_div(f.LM, g.LM)
if dv == 0 and monom is not None:
q = q.add_term(cf, monom)
f -= g.mul_term(cf, monom)
else:
r = r.add_term(*f.LT)
f = f.kill_lead_term()
return q, r
def combinations(lisp, m):
def recursion(fa, lisp, m):
if m == 0:
yield fa
else:
for i, fa2 in enumerate(lisp[0:len(lisp) + 1 - m]):
for el in recursion(zzx_mul(fa2, fa), list(lisp[i + 1:]),
m - 1):
yield el
for i, fa in enumerate(lisp[0:len(lisp) + 1 - m]):
for el in recursion(fa, list(lisp[i + 1:]), m - 1):
yield el
debug = flags.get('debug', False)
cont, f = f.as_primitive()
N = len(symbols)
max_exp = {}
for v in symbols:
max_exp[v] = 0
for coeff, monom in f.iter_terms():
for v, exp in zip(symbols, monom):
if exp > max_exp[v]:
max_exp[v] = exp
symbols = sorted(symbols, reverse=True, key=lambda v: max_exp[v])
f = Poly(f, *symbols)
d = max_exp[symbols[0]] + 1
terms, exps = {}, []
for i in xrange(0, len(symbols)):
exps.append(d**i)
for coeff, monom in f.iter_terms():
exp = 0
for i, expi in enumerate(monom):
exp += expi * exps[i]
terms[exp] = int(coeff)
g, factors = zzx_from_dict(terms), []
try:
for ff, k in zzx_factor(g)[1]:
for i in xrange(0, k):
factors.append(ff)
except OverflowError:
raise PolynomialError(
"input too large for multivariate Kronecker method")
const, result, tested = 1, [], []
if debug: print "KRONECKER-MV: Z[x] #factors = %i ..." % (len(factors))
for k in range(1, len(factors) // 2 + 1):
for h in combinations(factors, k):
if h in tested:
continue
n = zzx_degree(h)
terms = {}
for coeff in h:
if not coeff:
n = n - 1
continue
else:
coeff = Integer(coeff)
y_deg, n = n, n - 1
monom = [0] * N
for i in xrange(N):
v_deg = y_deg % d
y_deg = (y_deg - v_deg) // d
monom[i] = v_deg
monom = tuple(monom)
if terms.has_key(monom):
terms[monom] += coeff
else:
terms[monom] = coeff
cand = Poly(terms, *symbols)
if cand.is_one:
continue
if cand.LC.is_negative:
cand = -cand
q, r = mv_int_div(f, cand)
if r.is_zero:
if debug: print "KRONECKER-MV: Z[X] factor found %s" % cand
result.append(cand)
f = q
else:
tested.append(h)
if f.is_constant:
const, f = f.LC, Poly(1, *symbols)
break
if f.is_one:
break
if not f.is_one:
if debug: print "KRONECKER-MV: Z[X] factor found %s" % f
result.append(f)
factors = {}
for ff in result:
if factors.has_key(ff):
factors[ff] += 1
else:
factors[ff] = 1
return cont * const, sorted(factors.items())
def test_zzx_degree():
assert zzx_degree([]) == -1
assert zzx_degree([1]) == 0
assert zzx_degree([1, 0]) == 1
assert zzx_degree([1, 0, 0, 0, 1]) == 4
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 kronecker_mv(f, **flags):
"""Kronecker method for Z[X] polynomials.
NOTE: This function is very slow even on small input.
Use debug=True flag to see its progress, if any.
"""
symbols = f.symbols
def mv_int_div(f, g):
q = Poly((), *symbols)
r = Poly((), *symbols)
while not f.is_zero:
lc_f, lc_g = f.LC, g.LC
dv = lc_f % lc_g
cf = lc_f / lc_g
monom = monomial_div(f.LM, g.LM)
if dv == 0 and monom is not None:
q = q.add_term(cf, monom)
f -= g.mul_term(cf, monom)
else:
r = r.add_term(*f.LT)
f = f.kill_lead_term()
return q, r
def combinations(lisp, m):
def recursion(fa, lisp, m):
if m == 0:
yield fa
else:
for i, fa2 in enumerate(lisp[0 : len(lisp) + 1 - m]):
for el in recursion(zzx_mul(fa2, fa), list(lisp[i + 1:]), m - 1):
yield el
for i, fa in enumerate(lisp[0 : len(lisp) + 1 - m]):
for el in recursion(fa, list(lisp[i + 1:]), m - 1):
yield el
debug = flags.get('debug', False)
cont, f = f.as_primitive()
N = len(symbols)
max_exp = {}
for v in symbols:
max_exp[v] = 0
for coeff, monom in f.iter_terms():
for v, exp in zip(symbols, monom):
if exp > max_exp[v]:
max_exp[v] = exp
symbols = sorted(symbols, reverse=True,
key=lambda v: max_exp[v])
f = Poly(f, *symbols)
d = max_exp[symbols[0]] + 1
terms, exps = {}, []
for i in xrange(0, len(symbols)):
exps.append(d**i)
for coeff, monom in f.iter_terms():
exp = 0
for i, expi in enumerate(monom):
exp += expi * exps[i]
terms[exp] = int(coeff)
g, factors = zzx_from_dict(terms), []
try:
for ff, k in zzx_factor(g)[1]:
for i in xrange(0, k):
factors.append(ff)
except OverflowError:
raise PolynomialError("input too large for multivariate Kronecker method")
const, result, tested = 1, [], []
if debug: print "KRONECKER-MV: Z[x] #factors = %i ..." % (len(factors))
for k in range(1, len(factors)//2 + 1):
for h in combinations(factors, k):
if h in tested:
continue
n = zzx_degree(h)
terms = {}
for coeff in h:
if not coeff:
n = n-1
continue
else:
coeff = Integer(coeff)
y_deg, n = n, n-1
monom = [0] * N
for i in xrange(N):
v_deg = y_deg % d
y_deg = (y_deg - v_deg) // d
monom[i] = v_deg
monom = tuple(monom)
if terms.has_key(monom):
terms[monom] += coeff
else:
terms[monom] = coeff
cand = Poly(terms, *symbols)
if cand.is_one:
continue
if cand.LC.is_negative:
cand = -cand;
q, r = mv_int_div(f, cand)
if r.is_zero:
if debug: print "KRONECKER-MV: Z[X] factor found %s" % cand
result.append(cand)
f = q
else:
tested.append(h)
if f.is_constant:
const, f = f.LC, Poly(1, *symbols)
break
if f.is_one:
break
if not f.is_one:
if debug: print "KRONECKER-MV: Z[X] factor found %s" % f
result.append(f)
factors = {}
for ff in result:
if factors.has_key(ff):
factors[ff] += 1
else:
factors[ff] = 1
return cont*const, sorted(factors.items())
