def _roots_univariate_polynomial(self, p, ring=None, multiplicities=None, algorithm=None):
r"""
Return a list of pairs ``(root,multiplicity)`` of roots of the polynomial ``p``.
If the argument ``multiplicities`` is set to ``False`` then return the
list of roots.
.. SEEALSO::
:meth:`_factor_univariate_polynomial`
EXAMPLES::
sage: R.<x> = PolynomialRing(GF(5),'x')
sage: K = GF(5).algebraic_closure('t')
sage: sorted((x^6 - 1).roots(K,multiplicities=False))
[1, 4, 2*t2 + 1, 2*t2 + 2, 3*t2 + 3, 3*t2 + 4]
sage: ((K.gen(2)*x - K.gen(3))**2).roots(K)
[(3*t6^5 + 2*t6^4 + 2*t6^2 + 3, 2)]
sage: for _ in range(10):
....: p = R.random_element(degree=randint(2,8))
....: for r in p.roots(K, multiplicities=False):
....: assert p(r).is_zero(), "r={} is not a root of p={}".format(r,p)
"""
from sage.arith.all import lcm
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
# first build a polynomial over some finite field
coeffs = [v.as_finite_field_element(minimal=True) for v in p.list()]
l = lcm([c[0].degree() for c in coeffs])
F, phi = self.subfield(l)
P = p.parent().change_ring(F)
new_coeffs = [self.inclusion(c[0].degree(), l)(c[1]) for c in coeffs]
polys = [(g,m,l,phi) for g,m in P(new_coeffs).factor()]
roots = [] # a list of pair (root,multiplicity)
while polys:
g,m,l,phi = polys.pop()
if g.degree() == 1: # found a root
r = phi(-g.constant_coefficient())
roots.append((r,m))
else: # look at the extension of degree g.degree() which contains at
# least one root of g
ll = l * g.degree()
psi = self.inclusion(l, ll)
FF, pphi = self.subfield(ll)
# note: there is no coercion from the l-th subfield to the ll-th
# subfield. The line below does the conversion manually.
g = PolynomialRing(FF, 'x')([psi(_) for _ in g])
polys.extend((gg,m,ll,pphi) for gg,_ in g.factor())
if multiplicities:
return roots
else:
return [r[0] for r in roots]
def _roots_univariate_polynomial(self, p, ring=None, multiplicities=None, algorithm=None):
r"""
Return a list of pairs ``(root,multiplicity)`` of roots of the polynomial ``p``.
If the argument ``multiplicities`` is set to ``False`` then return the
list of roots.
.. SEEALSO::
:meth:`_factor_univariate_polynomial`
EXAMPLES::
sage: R.<x> = PolynomialRing(GF(5),'x')
sage: K = GF(5).algebraic_closure('t')
sage: sorted((x^6 - 1).roots(K,multiplicities=False))
[1, 4, 2*t2 + 1, 2*t2 + 2, 3*t2 + 3, 3*t2 + 4]
sage: ((K.gen(2)*x - K.gen(3))**2).roots(K)
[(3*t6^5 + 2*t6^4 + 2*t6^2 + 3, 2)]
sage: for _ in range(10):
....: p = R.random_element(degree=randint(2,8))
....: for r in p.roots(K, multiplicities=False):
....: assert p(r).is_zero(), "r={} is not a root of p={}".format(r,p)
"""
from sage.arith.all import lcm
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
# first build a polynomial over some finite field
coeffs = [v.as_finite_field_element(minimal=True) for v in p.list()]
l = lcm([c[0].degree() for c in coeffs])
F, phi = self.subfield(l)
P = p.parent().change_ring(F)
new_coeffs = [self.inclusion(c[0].degree(), l)(c[1]) for c in coeffs]
polys = [(g,m,l,phi) for g,m in P(new_coeffs).factor()]
roots = [] # a list of pair (root,multiplicity)
while polys:
g,m,l,phi = polys.pop()
if g.degree() == 1: # found a root
r = phi(-g.constant_coefficient())
roots.append((r,m))
else: # look at the extension of degree g.degree() which contains at
# least one root of g
ll = l * g.degree()
psi = self.inclusion(l, ll)
FF, pphi = self.subfield(ll)
# note: there is no coercion from the l-th subfield to the ll-th
# subfield. The line below does the conversion manually.
g = PolynomialRing(FF, 'x')([psi(_) for _ in g])
polys.extend((gg,m,ll,pphi) for gg,_ in g.factor())
if multiplicities:
return roots
else:
return [r[0] for r in roots]
