def dmp_resultant(f, g, u, K, includePRS=False):
"""
Computes resultant of two polynomials in `K[X]`.
Examples
========
>>> from diofant.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> f = 3*x**2*y - y**3 - 4
>>> g = x**2 + x*y**3 - 9
>>> R.dmp_resultant(f, g)
-3*y**10 - 12*y**7 + y**6 - 54*y**4 + 8*y**3 + 729*y**2 - 216*y + 16
"""
if not u:
return dup_resultant(f, g, K, includePRS=includePRS)
if includePRS:
return dmp_prs_resultant(f, g, u, K)
if K.has_Field:
if K.is_QQ and query('USE_COLLINS_RESULTANT'):
return dmp_qq_collins_resultant(f, g, u, K)
else:
if K.is_ZZ and query('USE_COLLINS_RESULTANT'):
return dmp_zz_collins_resultant(f, g, u, K)
return dmp_prs_resultant(f, g, u, K)[0]
def dup_zz_factor_sqf(f, K):
"""Factor square-free (non-primitive) polyomials in `Z[x]`. """
cont, g = dup_primitive(f, K)
n = dup_degree(g)
if dup_LC(g, K) < 0:
cont, g = -cont, dup_neg(g, K)
if n <= 0:
return cont, []
elif n == 1:
return cont, [g]
if query('USE_IRREDUCIBLE_IN_FACTOR'):
if dup_zz_irreducible_p(g, K):
return cont, [g]
factors = None
if query('USE_CYCLOTOMIC_FACTOR'):
factors = dup_zz_cyclotomic_factor(g, K)
if factors is None:
factors = dup_zz_zassenhaus(g, K)
return cont, _sort_factors(factors, multiple=False)
def _dmp_inner_gcd(f, g, u, K):
"""Helper function for `dmp_inner_gcd()`. """
if not K.is_Exact:
try:
exact = K.get_exact()
except DomainError:
return dmp_one(u, K), f, g
f = dmp_convert(f, u, K, exact)
g = dmp_convert(g, u, K, exact)
h, cff, cfg = _dmp_inner_gcd(f, g, u, exact)
h = dmp_convert(h, u, exact, K)
cff = dmp_convert(cff, u, exact, K)
cfg = dmp_convert(cfg, u, exact, K)
return h, cff, cfg
elif K.has_Field:
if K.is_QQ and query('USE_HEU_GCD'):
try:
return dmp_qq_heu_gcd(f, g, u, K)
except HeuristicGCDFailed:
pass
return dmp_ff_prs_gcd(f, g, u, K)
else:
if K.is_ZZ and query('USE_HEU_GCD'):
try:
return dmp_zz_heu_gcd(f, g, u, K)
except HeuristicGCDFailed:
pass
return dmp_rr_prs_gcd(f, g, u, K)
def test_configure(monkeypatch):
assert query('heu_gcd_max') == 6
monkeypatch.setenv('DIOFANT_HEU_GCD_MAX', 7)
configure()
assert query('heu_gcd_max') == 7
monkeypatch.setenv('DIOFANT_HEU_GCD_MAX', '1^2')
configure()
assert query('heu_gcd_max') == 6
monkeypatch.setenv('DIOFANT_HEU_GCD_MAX', '"abcd"')
configure()
assert query('heu_gcd_max') == 6
def test_configure(monkeypatch):
assert query('heu_gcd_max') == 6
monkeypatch.setenv('DIOFANT_HEU_GCD_MAX', '7')
configure()
assert query('heu_gcd_max') == 7
monkeypatch.setenv('DIOFANT_HEU_GCD_MAX', '1^2')
configure()
assert query('heu_gcd_max') == 6
monkeypatch.setenv('DIOFANT_HEU_GCD_MAX', '"abcd"')
configure()
assert query('heu_gcd_max') == 6
def dup_inner_gcd(f, g, K):
"""
Computes polynomial GCD and cofactors of `f` and `g` in `K[x]`.
Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``,
``cff = quo(f, h)``, and ``cfg = quo(g, h)``.
Examples
========
>>> from diofant.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_inner_gcd(x**2 - 1, x**2 - 3*x + 2)
(x - 1, x + 1, x - 2)
"""
if not K.is_Exact:
try:
exact = K.get_exact()
except DomainError:
return [K.one], f, g
f = dup_convert(f, K, exact)
g = dup_convert(g, K, exact)
h, cff, cfg = dup_inner_gcd(f, g, exact)
h = dup_convert(h, exact, K)
cff = dup_convert(cff, exact, K)
cfg = dup_convert(cfg, exact, K)
return h, cff, cfg
elif K.has_Field:
if K.is_QQ and query('USE_HEU_GCD'):
try:
return dup_qq_heu_gcd(f, g, K)
except HeuristicGCDFailed:
pass
return dup_ff_prs_gcd(f, g, K)
else:
if K.is_ZZ and query('USE_HEU_GCD'):
try:
return dup_zz_heu_gcd(f, g, K)
except HeuristicGCDFailed:
pass
return dup_rr_prs_gcd(f, g, K)
def gf_factor_sqf(f, p, K, method=None):
"""
Factor a square-free polynomial ``f`` in ``GF(p)[x]``.
Examples
========
>>> from diofant.polys.domains import ZZ
>>> gf_factor_sqf(ZZ.map([3, 2, 4]), 5, ZZ)
(3, [[1, 1], [1, 3]])
"""
lc, f = gf_monic(f, p, K)
if gf_degree(f) < 1:
return lc, []
method = method or query('GF_FACTOR_METHOD')
if method is not None:
factors = _factor_methods[method](f, p, K)
else:
factors = gf_zassenhaus(f, p, K)
return lc, factors
def _dmp_ff_trivial_gcd(f, g, u, K):
"""Handle trivial cases in GCD algorithm over a field. """
zero_f = dmp_zero_p(f, u)
zero_g = dmp_zero_p(g, u)
if zero_f and zero_g:
return tuple(dmp_zeros(3, u, K))
elif zero_f:
return (dmp_ground_monic(g, u, K), dmp_zero(u),
dmp_ground(dmp_ground_LC(g, u, K), u))
elif zero_g:
return (dmp_ground_monic(f, u, K), dmp_ground(dmp_ground_LC(f, u, K),
u), dmp_zero(u))
elif query('USE_SIMPLIFY_GCD'):
return _dmp_simplify_gcd(f, g, u, K)
def _dmp_rr_trivial_gcd(f, g, u, K):
"""Handle trivial cases in GCD algorithm over a ring. """
zero_f = dmp_zero_p(f, u)
zero_g = dmp_zero_p(g, u)
if zero_f and zero_g:
return tuple(dmp_zeros(3, u, K))
elif zero_f:
if K.is_nonnegative(dmp_ground_LC(g, u, K)):
return g, dmp_zero(u), dmp_one(u, K)
else:
return dmp_neg(g, u, K), dmp_zero(u), dmp_ground(-K.one, u)
elif zero_g:
if K.is_nonnegative(dmp_ground_LC(f, u, K)):
return f, dmp_one(u, K), dmp_zero(u)
else:
return dmp_neg(f, u, K), dmp_ground(-K.one, u), dmp_zero(u)
elif dmp_one_p(f, u, K) or dmp_one_p(g, u, K):
return dmp_one(u, K), f, g
elif query('USE_SIMPLIFY_GCD'):
return _dmp_simplify_gcd(f, g, u, K)
def gf_irreducible_p(f, p, K):
"""
Test irreducibility of a polynomial ``f`` in ``GF(p)[x]``.
Examples
========
>>> from diofant.polys.domains import ZZ
>>> gf_irreducible_p(ZZ.map([1, 4, 2, 2, 3, 2, 4, 1, 4, 0, 4]), 5, ZZ)
True
>>> gf_irreducible_p(ZZ.map([3, 2, 4]), 5, ZZ)
False
"""
method = query('GF_IRRED_METHOD')
if method is not None:
irred = _irred_methods[method](f, p, K)
else:
irred = gf_irred_p_rabin(f, p, K)
return irred
def test_configure2():
configure()
assert query('heu_gcd_max') == 6
def test_configure2():
configure()
assert query('heu_gcd_max') == 6
def dmp_zz_wang(f, u, K, mod=None, seed=None):
"""
Factor primitive square-free polynomials in `Z[X]`.
Given a multivariate polynomial `f` in `Z[x_1,...,x_n]`, which is
primitive and square-free in `x_1`, computes factorization of `f` into
irreducibles over integers.
The procedure is based on Wang's Enhanced Extended Zassenhaus
algorithm. The algorithm works by viewing `f` as a univariate polynomial
in `Z[x_2,...,x_n][x_1]`, for which an evaluation mapping is computed::
x_2 -> a_2, ..., x_n -> a_n
where `a_i`, for `i = 2, ..., n`, are carefully chosen integers. The
mapping is used to transform `f` into a univariate polynomial in `Z[x_1]`,
which can be factored efficiently using Zassenhaus algorithm. The last
step is to lift univariate factors to obtain true multivariate
factors. For this purpose a parallel Hensel lifting procedure is used.
The parameter ``seed`` is passed to _randint and can be used to seed randint
(when an integer) or (for testing purposes) can be a sequence of numbers.
References
==========
.. [1] [Wang78]_
.. [2] [Geddes92]_
"""
from diofant.utilities.randtest import _randint
randint = _randint(seed)
ct, T = dmp_zz_factor(dmp_LC(f, K), u - 1, K)
b = dmp_zz_mignotte_bound(f, u, K)
p = K(nextprime(b))
if mod is None:
if u == 1:
mod = 2
else:
mod = 1
history, configs, A, r = set(), [], [K.zero] * u, None
try:
cs, s, E = dmp_zz_wang_test_points(f, T, ct, A, u, K)
_, H = dup_zz_factor_sqf(s, K)
r = len(H)
if r == 1:
return [f]
configs = [(s, cs, E, H, A)]
except EvaluationFailed:
pass
eez_num_configs = query('EEZ_NUMBER_OF_CONFIGS')
eez_num_tries = query('EEZ_NUMBER_OF_TRIES')
eez_mod_step = query('EEZ_MODULUS_STEP')
while len(configs) < eez_num_configs:
for _ in range(eez_num_tries):
A = [K(randint(-mod, mod)) for _ in range(u)]
if tuple(A) not in history:
history.add(tuple(A))
else:
continue
try:
cs, s, E = dmp_zz_wang_test_points(f, T, ct, A, u, K)
except EvaluationFailed:
continue
_, H = dup_zz_factor_sqf(s, K)
rr = len(H)
if r is not None:
if rr != r: # pragma: no cover
if rr < r:
configs, r = [], rr
else:
continue
else:
r = rr
if r == 1:
return [f]
configs.append((s, cs, E, H, A))
if len(configs) == eez_num_configs:
break
else:
mod += eez_mod_step
s_norm, s_arg, i = None, 0, 0
for s, _, _, _, _ in configs:
_s_norm = dup_max_norm(s, K)
if s_norm is not None:
if _s_norm < s_norm:
s_norm = _s_norm
s_arg = i
else:
s_norm = _s_norm
i += 1
_, cs, E, H, A = configs[s_arg]
orig_f = f
try:
f, H, LC = dmp_zz_wang_lead_coeffs(f, T, cs, E, H, A, u, K)
factors = dmp_zz_wang_hensel_lifting(f, H, LC, A, p, u, K)
except ExtraneousFactors: # pragma: no cover
if query('EEZ_RESTART_IF_NEEDED'):
return dmp_zz_wang(orig_f, u, K, mod + 1)
else:
raise ExtraneousFactors(
"we need to restart algorithm with better parameters")
negative, result = 0, []
for f in factors:
_, f = dmp_ground_primitive(f, u, K)
if K.is_negative(dmp_ground_LC(f, u, K)):
f = dmp_neg(f, u, K)
result.append(f)
return result
def dup_zz_factor(f, K):
"""
Factor (non square-free) polynomials in `Z[x]`.
Given a univariate polynomial `f` in `Z[x]` computes its complete
factorization `f_1, ..., f_n` into irreducibles over integers::
f = content(f) f_1**k_1 ... f_n**k_n
The factorization is computed by reducing the input polynomial
into a primitive square-free polynomial and factoring it using
Zassenhaus algorithm. Trial division is used to recover the
multiplicities of factors.
The result is returned as a tuple consisting of::
(content(f), [(f_1, k_1), ..., (f_n, k_n))
Consider polynomial `f = 2*x**4 - 2`::
>>> from diofant.polys import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_zz_factor(2*x**4 - 2)
(2, [(x - 1, 1), (x + 1, 1), (x**2 + 1, 1)])
In result we got the following factorization::
f = 2 (x - 1) (x + 1) (x**2 + 1)
Note that this is a complete factorization over integers,
however over Gaussian integers we can factor the last term.
By default, polynomials `x**n - 1` and `x**n + 1` are factored
using cyclotomic decomposition to speedup computations. To
disable this behaviour set cyclotomic=False.
References
==========
.. [1] [Gathen99]_
"""
cont, g = dup_primitive(f, K)
n = dup_degree(g)
if dup_LC(g, K) < 0:
cont, g = -cont, dup_neg(g, K)
if n <= 0:
return cont, []
elif n == 1:
return cont, [(g, 1)]
if query('USE_IRREDUCIBLE_IN_FACTOR'):
if dup_zz_irreducible_p(g, K):
return cont, [(g, 1)]
g = dup_sqf_part(g, K)
H = None
if query('USE_CYCLOTOMIC_FACTOR'):
H = dup_zz_cyclotomic_factor(g, K)
if H is None:
H = dup_zz_zassenhaus(g, K)
factors = dup_trial_division(f, H, K)
return cont, factors