def dmp_sqf_list(f, u, K, all=False):
"""
Return square-free decomposition of a polynomial in ``K[X]``.
Examples
========
>>> from diofant.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 test_dmp_diff_in():
assert dmp_diff_in(f_6, 2, 1, 3, ZZ) == \
dmp_swap(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 2, 3, ZZ), 0, 1, 3, ZZ)
assert dmp_diff_in(f_6, 3, 1, 3, ZZ) == \
dmp_swap(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 3, 3, ZZ), 0, 1, 3, ZZ)
assert dmp_diff_in(f_6, 2, 2, 3, ZZ) == \
dmp_swap(dmp_diff(dmp_swap(f_6, 0, 2, 3, ZZ), 2, 3, ZZ), 0, 2, 3, ZZ)
assert dmp_diff_in(f_6, 3, 2, 3, ZZ) == \
dmp_swap(dmp_diff(dmp_swap(f_6, 0, 2, 3, ZZ), 3, 3, ZZ), 0, 2, 3, ZZ)
def test_dmp_diff_in():
assert dmp_diff_in(f_6, 2, 1, 3, ZZ) == \
dmp_swap(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 2, 3, ZZ), 0, 1, 3, ZZ)
assert dmp_diff_in(f_6, 3, 1, 3, ZZ) == \
dmp_swap(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 3, 3, ZZ), 0, 1, 3, ZZ)
assert dmp_diff_in(f_6, 2, 2, 3, ZZ) == \
dmp_swap(dmp_diff(dmp_swap(f_6, 0, 2, 3, ZZ), 2, 3, ZZ), 0, 2, 3, ZZ)
assert dmp_diff_in(f_6, 3, 2, 3, ZZ) == \
dmp_swap(dmp_diff(dmp_swap(f_6, 0, 2, 3, ZZ), 3, 3, ZZ), 0, 2, 3, ZZ)
pytest.raises(IndexError, lambda: dmp_diff_in(f_6, 2, -1, 3, ZZ))
pytest.raises(IndexError, lambda: dmp_diff_in(f_6, 2, 1, 0, ZZ))
def dmp_sqf_part(f, u, K):
"""
Returns square-free part of a polynomial in ``K[X]``.
Examples
========
>>> from diofant.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> R.dmp_sqf_part(x**3 + 2*x**2*y + x*y**2)
x**2 + x*y
"""
if not u:
return dup_sqf_part(f, K)
if K.is_FiniteField:
return dmp_gf_sqf_part(f, u, K)
if dmp_zero_p(f, u):
return f
if K.is_negative(dmp_ground_LC(f, u, K)):
f = dmp_neg(f, u, K)
gcd = dmp_gcd(f, dmp_diff(f, 1, u, K), u, K)
sqf = dmp_quo(f, gcd, u, K)
if K.has_Field:
return dmp_ground_monic(sqf, u, K)
else:
return dmp_ground_primitive(sqf, u, K)[1]
def dmp_discriminant(f, u, K):
"""
Computes discriminant of a polynomial in `K[X]`.
Examples
========
>>> from diofant.polys import ring, ZZ
>>> R, x,y,z,t = ring("x,y,z,t", ZZ)
>>> R.dmp_discriminant(x**2*y + x*z + t)
-4*y*t + z**2
"""
if not u:
return dup_discriminant(f, K)
d, v = dmp_degree(f, u), u - 1
if d <= 0:
return dmp_zero(v)
else:
s = (-1)**((d * (d - 1)) // 2)
c = dmp_LC(f, K)
r = dmp_resultant(f, dmp_diff(f, 1, u, K), u, K)
c = dmp_mul_ground(c, K(s), v, K)
return dmp_quo(r, c, v, K)
def dmp_sqf_p(f, u, K):
"""
Return ``True`` if ``f`` is a square-free polynomial in ``K[X]``.
Examples
========
>>> from diofant.polys import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> R.dmp_sqf_p(x**2 + 2*x*y + y**2)
False
>>> R.dmp_sqf_p(x**2 + y**2)
True
"""
if dmp_zero_p(f, u):
return True
else:
return not dmp_degree(dmp_gcd(f, dmp_diff(f, 1, u, K), u, K), u)
def test_dmp_diff_eval_in():
assert dmp_diff_eval_in(f_6, 2, 7, 1, 3, ZZ) == \
dmp_eval(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 2, 3, ZZ), 7, 3, ZZ)
pytest.raises(IndexError, lambda: dmp_diff_eval_in(f_6, 2, 7, 4, 3, ZZ))
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 test_dmp_diff_eval_in():
assert dmp_diff_eval_in(f_6, 2, 7, 1, 3, ZZ) == \
dmp_eval(dmp_diff(dmp_swap(f_6, 0, 1, 3, ZZ), 2, 3, ZZ), 7, 3, ZZ)