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"""
Polynomial Template for C/C++ Library Interfaces
"""
#*****************************************************************************
# Copyright (C) 2008 Martin Albrecht <M.R.Albrecht@rhul.ac.uk>
# Copyright (C) 2008 Robert Bradshaw
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
# http://www.gnu.org/licenses/
#*****************************************************************************
from sage.rings.polynomial.polynomial_element cimport Polynomial
from sage.structure.element cimport ModuleElement, Element, RingElement
from sage.structure.element import coerce_binop, bin_op
from sage.structure.richcmp cimport rich_to_bool
from sage.rings.fraction_field_element import FractionFieldElement
from sage.rings.integer cimport Integer
from sage.libs.all import pari_gen
import operator
from sage.interfaces.all import singular as singular_default
def make_element(parent, args):
return parent(*args)
cdef inline Polynomial_template element_shift(self, int n):
if not isinstance(self, Polynomial_template):
if n > 0:
error_msg = "Cannot shift %s << %n."%(self, n)
else:
error_msg = "Cannot shift %s >> %n."%(self, n)
raise TypeError(error_msg)
if n == 0:
return self
cdef celement *gen = celement_new((<Polynomial_template>self)._cparent)
celement_gen(gen, 0, (<Polynomial_template>self)._cparent)
celement_pow(gen, gen, abs(n), NULL, (<Polynomial_template>self)._cparent)
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
if n > 0:
celement_mul(&r.x, &(<Polynomial_template>self).x, gen, (<Polynomial_template>self)._cparent)
else:
celement_floordiv(&r.x, &(<Polynomial_template>self).x, gen, (<Polynomial_template>self)._cparent)
celement_delete(gen, (<Polynomial_template>self)._cparent)
return r
cdef class Polynomial_template(Polynomial):
r"""
Template for interfacing to external C / C++ libraries for implementations of polynomials.
AUTHORS:
- Robert Bradshaw (2008-10): original idea for templating
- Martin Albrecht (2008-10): initial implementation
This file implements a simple templating engine for linking univariate
polynomials to their C/C++ library implementations. It requires a
'linkage' file which implements the ``celement_`` functions (see
:mod:`sage.libs.ntl.ntl_GF2X_linkage` for an example). Both parts are
then plugged together by inclusion of the linkage file when inheriting from
this class. See :mod:`sage.rings.polynomial.polynomial_gf2x` for an
example.
We illustrate the generic glueing using univariate polynomials over
`\mathop{\mathrm{GF}}(2)`.
.. note::
Implementations using this template MUST implement coercion from base
ring elements and :meth:`get_unsafe`. See
:class:`~sage.rings.polynomial.polynomial_gf2x.Polynomial_GF2X` for an
example.
"""
def __init__(self, parent, x=None, check=True, is_gen=False, construct=False):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: P(0)
0
sage: P(GF(2)(1))
1
sage: P(3)
1
sage: P([1,0,1])
x^2 + 1
sage: P(list(map(GF(2),[1,0,1])))
x^2 + 1
"""
cdef celement *gen
cdef celement *monomial
cdef Py_ssize_t deg
Polynomial.__init__(self, parent, is_gen=is_gen)
(<Polynomial_template>self)._cparent = get_cparent(self._parent)
if is_gen:
celement_construct(&self.x, (<Polynomial_template>self)._cparent)
celement_gen(&self.x, 0, (<Polynomial_template>self)._cparent)
elif isinstance(x, Polynomial_template):
try:
celement_construct(&self.x, (<Polynomial_template>self)._cparent)
celement_set(&self.x, &(<Polynomial_template>x).x, (<Polynomial_template>self)._cparent)
except NotImplementedError:
raise TypeError("%s not understood."%x)
elif isinstance(x, int) or isinstance(x, Integer):
try:
celement_construct(&self.x, (<Polynomial_template>self)._cparent)
celement_set_si(&self.x, int(x), (<Polynomial_template>self)._cparent)
except NotImplementedError:
raise TypeError("%s not understood."%x)
elif isinstance(x, list) or isinstance(x, tuple):
celement_construct(&self.x, (<Polynomial_template>self)._cparent)
gen = celement_new((<Polynomial_template>self)._cparent)
monomial = celement_new((<Polynomial_template>self)._cparent)
celement_set_si(&self.x, 0, (<Polynomial_template>self)._cparent)
celement_gen(gen, 0, (<Polynomial_template>self)._cparent)
deg = 0
for e in x:
# r += parent(e)*power
celement_pow(monomial, gen, deg, NULL, (<Polynomial_template>self)._cparent)
celement_mul(monomial, &(<Polynomial_template>self.__class__(parent, e)).x, monomial, (<Polynomial_template>self)._cparent)
celement_add(&self.x, &self.x, monomial, (<Polynomial_template>self)._cparent)
deg += 1
celement_delete(gen, (<Polynomial_template>self)._cparent)
celement_delete(monomial, (<Polynomial_template>self)._cparent)
elif isinstance(x, dict):
celement_construct(&self.x, (<Polynomial_template>self)._cparent)
gen = celement_new((<Polynomial_template>self)._cparent)
monomial = celement_new((<Polynomial_template>self)._cparent)
celement_set_si(&self.x, 0, (<Polynomial_template>self)._cparent)
celement_gen(gen, 0, (<Polynomial_template>self)._cparent)
for deg, coef in x.iteritems():
celement_pow(monomial, gen, deg, NULL, (<Polynomial_template>self)._cparent)
celement_mul(monomial, &(<Polynomial_template>self.__class__(parent, coef)).x, monomial, (<Polynomial_template>self)._cparent)
celement_add(&self.x, &self.x, monomial, (<Polynomial_template>self)._cparent)
celement_delete(gen, (<Polynomial_template>self)._cparent)
celement_delete(monomial, (<Polynomial_template>self)._cparent)
elif isinstance(x, pari_gen):
k = (<Polynomial_template>self)._parent.base_ring()
x = [k(w) for w in x.list()]
self.__class__.__init__(self, parent, x, check=True, is_gen=False, construct=construct)
elif isinstance(x, Polynomial):
k = (<Polynomial_template>self)._parent.base_ring()
x = [k(w) for w in list(x)]
Polynomial_template.__init__(self, parent, x, check=True, is_gen=False, construct=construct)
elif isinstance(x, FractionFieldElement) and (x.parent().base() is parent or x.parent().base() == parent) and x.denominator() == 1:
x = x.numerator()
self.__class__.__init__(self, parent, x, check=check, is_gen=is_gen, construct=construct)
else:
x = parent.base_ring()(x)
self.__class__.__init__(self, parent, x, check=check, is_gen=is_gen, construct=construct)
def get_cparent(self):
return <long> self._cparent
def __reduce__(self):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: loads(dumps(x)) == x
True
"""
return make_element, ((<Polynomial_template>self)._parent, (self.list(), False, self.is_gen()))
cpdef list list(self, bint copy=True):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: x.list()
[0, 1]
sage: list(x)
[0, 1]
"""
cdef Py_ssize_t i
return [self[i] for i in range(celement_len(&self.x, (<Polynomial_template>self)._cparent))]
def __dealloc__(self):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: del x
TESTS:
The following has been a problem in a preliminary version of
:trac:`12313`::
sage: K.<z> = GF(4)
sage: P.<x> = K[]
sage: del P
sage: del x
sage: import gc
sage: _ = gc.collect()
"""
celement_destruct(&self.x, (<Polynomial_template>self)._cparent)
cpdef _add_(self, right):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: x + 1
x + 1
"""
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
celement_add(&r.x, &(<Polynomial_template>self).x, &(<Polynomial_template>right).x, (<Polynomial_template>self)._cparent)
#assert(r._parent(pari(self) + pari(right)) == r)
return r
cpdef _sub_(self, right):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: x - 1
x + 1
"""
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
celement_sub(&r.x, &(<Polynomial_template>self).x, &(<Polynomial_template>right).x, (<Polynomial_template>self)._cparent)
#assert(r._parent(pari(self) - pari(right)) == r)
return r
def __neg__(self):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: -x
x
"""
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
celement_neg(&r.x, &self.x, (<Polynomial_template>self)._cparent)
#assert(r._parent(-pari(self)) == r)
return r
cpdef _lmul_(self, Element left):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: t = x^2 + x + 1
sage: 0*t
0
sage: 1*t
x^2 + x + 1
sage: R.<y> = GF(5)[]
sage: u = y^2 + y + 1
sage: 3*u
3*y^2 + 3*y + 3
sage: 5*u
0
sage: (2^81)*u
2*y^2 + 2*y + 2
sage: (-2^81)*u
3*y^2 + 3*y + 3
::
sage: P.<x> = GF(2)[]
sage: t = x^2 + x + 1
sage: t*0
0
sage: t*1
x^2 + x + 1
sage: R.<y> = GF(5)[]
sage: u = y^2 + y + 1
sage: u*3
3*y^2 + 3*y + 3
sage: u*5
0
"""
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
celement_mul_scalar(&r.x, &(<Polynomial_template>self).x, left, (<Polynomial_template>self)._cparent)
return r
cpdef _mul_(self, right):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: x*(x+1)
x^2 + x
"""
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
celement_mul(&r.x, &(<Polynomial_template>self).x, &(<Polynomial_template>right).x, (<Polynomial_template>self)._cparent)
#assert(r._parent(pari(self) * pari(right)) == r)
return r
@coerce_binop
def gcd(self, Polynomial_template other):
"""
Return the greatest common divisor of self and other.
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: f = x*(x+1)
sage: f.gcd(x+1)
x + 1
sage: f.gcd(x^2)
x
"""
if(celement_is_zero(&self.x, (<Polynomial_template>self)._cparent)):
return other
if(celement_is_zero(&other.x, (<Polynomial_template>self)._cparent)):
return self
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
celement_gcd(&r.x, &(<Polynomial_template>self).x, &(<Polynomial_template>other).x, (<Polynomial_template>self)._cparent)
#assert(r._parent(pari(self).gcd(pari(other))) == r)
return r
@coerce_binop
def xgcd(self, Polynomial_template other):
"""
Computes extended gcd of self and other.
EXAMPLES::
sage: P.<x> = GF(7)[]
sage: f = x*(x+1)
sage: f.xgcd(x+1)
(x + 1, 0, 1)
sage: f.xgcd(x^2)
(x, 1, 6)
"""
if(celement_is_zero(&self.x, (<Polynomial_template>self)._cparent)):
return other, self._parent(0), self._parent(1)
if(celement_is_zero(&other.x, (<Polynomial_template>self)._cparent)):
return self, self._parent(1), self._parent(0)
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
cdef Polynomial_template s = <Polynomial_template>T.__new__(T)
celement_construct(&s.x, (<Polynomial_template>self)._cparent)
s._parent = (<Polynomial_template>self)._parent
s._cparent = (<Polynomial_template>self)._cparent
cdef Polynomial_template t = <Polynomial_template>T.__new__(T)
celement_construct(&t.x, (<Polynomial_template>self)._cparent)
t._parent = (<Polynomial_template>self)._parent
t._cparent = (<Polynomial_template>self)._cparent
celement_xgcd(&r.x, &s.x, &t.x, &(<Polynomial_template>self).x, &(<Polynomial_template>other).x, (<Polynomial_template>self)._cparent)
#rp, sp, tp = pari(self).xgcd(pari(other))
#assert(r._parent(rp) == r)
#assert(s._parent(sp) == s)
#assert(t._parent(tp) == t)
return r,s,t
cpdef _floordiv_(self, right):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: x//(x + 1)
1
sage: (x + 1)//x
1
sage: F = GF(47)
sage: R.<x> = F[]
sage: x // 1
x
sage: x // F(1)
x
sage: 1 // x
0
sage: parent(x // 1)
Univariate Polynomial Ring in x over Finite Field of size 47
sage: parent(1 // x)
Univariate Polynomial Ring in x over Finite Field of size 47
"""
cdef Polynomial_template _right = <Polynomial_template>right
if celement_is_zero(&_right.x, (<Polynomial_template>self)._cparent):
raise ZeroDivisionError
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
#assert(r._parent(pari(self) // pari(right)) == r)
celement_floordiv(&r.x, &(<Polynomial_template>self).x, &(<Polynomial_template>right).x, (<Polynomial_template>self)._cparent)
return r
cpdef _mod_(self, other):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: (x^2 + 1) % x^2
1
TESTS::
We test that :trac:`10578` is fixed::
sage: P.<x> = GF(2)[]
sage: x % 1r
0
"""
cdef Polynomial_template _other = <Polynomial_template>other
if celement_is_zero(&_other.x, (<Polynomial_template>self)._cparent):
raise ZeroDivisionError
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
celement_mod(&r.x, &(<Polynomial_template>self).x, &_other.x, (<Polynomial_template>self)._cparent)
#assert(r._parent(pari(self) % pari(other)) == r)
return r
@coerce_binop
def quo_rem(self, Polynomial_template right):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: f = x^2 + x + 1
sage: f.quo_rem(x + 1)
(x, 1)
"""
if celement_is_zero(&right.x, (<Polynomial_template>self)._cparent):
raise ZeroDivisionError
cdef type T = type(self)
cdef Polynomial_template q = <Polynomial_template>T.__new__(T)
celement_construct(&q.x, (<Polynomial_template>self)._cparent)
q._parent = (<Polynomial_template>self)._parent
q._cparent = (<Polynomial_template>self)._cparent
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
celement_quorem(&q.x, &r.x, &(<Polynomial_template>self).x, &right.x, (<Polynomial_template>self)._cparent)
return q,r
def __long__(self):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: int(x)
Traceback (most recent call last):
...
ValueError: Cannot coerce polynomial with degree 1 to integer.
sage: int(P(1))
1
"""
if celement_len(&self.x, (<Polynomial_template>self)._cparent) > 1:
raise ValueError("Cannot coerce polynomial with degree %d to integer."%(self.degree()))
return int(self[0])
def __nonzero__(self):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: bool(x), x.is_zero()
(True, False)
sage: bool(P(0)), P(0).is_zero()
(False, True)
"""
return not celement_is_zero(&self.x, (<Polynomial_template>self)._cparent)
cpdef _richcmp_(self, other, int op):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: x != 1
True
sage: x < 1
False
sage: x > 1
True
"""
cdef int c
c = celement_cmp(&self.x, &(<Polynomial_template>other).x, self._cparent)
return rich_to_bool(op, c)
def __hash__(self):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: {x:1}
{x: 1}
"""
cdef long result = 0 # store it in a c-int and just let the overflowing additions wrap
cdef long result_mon
cdef long c_hash
cdef long var_name_hash
cdef int i
for i from 0<= i <= self.degree():
if i == 1:
# we delay the hashing until now to not waste it one a constant poly
var_name_hash = hash(self.variable_name())
# I'm assuming (incorrectly) that hashes of zero indicate that the element is 0.
# This assumption is not true, but I think it is true enough for the purposes and it
# it allows us to write fast code that omits terms with 0 coefficients. This is
# important if we want to maintain the '==' relationship with sparse polys.
c_hash = hash(self[i])
if c_hash != 0:
if i == 0:
result += c_hash
else:
# Hash (self[i], generator, i) as a tuple according to the algorithm.
result_mon = c_hash
result_mon = (1000003 * result_mon) ^ var_name_hash
result_mon = (1000003 * result_mon) ^ i
result += result_mon
if result == -1:
return -2
return result
def __pow__(self, ee, modulus):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: P.<x> = GF(2)[]
sage: x^1000
x^1000
sage: (x+1)^2
x^2 + 1
sage: (x+1)^(-2)
1/(x^2 + 1)
sage: f = x^9 + x^7 + x^6 + x^5 + x^4 + x^2 + x
sage: h = x^10 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1
sage: (f^2) % h
x^9 + x^8 + x^7 + x^5 + x^3
sage: pow(f, 2, h)
x^9 + x^8 + x^7 + x^5 + x^3
"""
if not isinstance(self, Polynomial_template):
raise NotImplementedError("%s^%s not defined."%(ee,self))
cdef bint recip = 0, do_sig
cdef long e
try:
e = ee
except OverflowError:
return Polynomial.__pow__(self, ee, modulus)
if e != ee:
raise TypeError("Only integral powers defined.")
elif e < 0:
recip = 1 # delay because powering frac field elements is slow
e = -e
if not self:
if e == 0:
raise ArithmeticError("0^0 is undefined.")
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
parent = (<Polynomial_template>self)._parent
r._parent = parent
r._cparent = (<Polynomial_template>self)._cparent
if modulus is None:
celement_pow(&r.x, &(<Polynomial_template>self).x, e, NULL, (<Polynomial_template>self)._cparent)
else:
if parent is not (<Polynomial_template>modulus)._parent and parent != (<Polynomial_template>modulus)._parent:
modulus = parent._coerce_(modulus)
celement_pow(&r.x, &(<Polynomial_template>self).x, e, &(<Polynomial_template>modulus).x, (<Polynomial_template>self)._cparent)
#assert(r._parent(pari(self)**ee) == r)
if recip:
return ~r
else:
return r
def __copy__(self):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: copy(x) is x
False
sage: copy(x) == x
True
"""
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
celement_set(&r.x, &self.x, (<Polynomial_template>self)._cparent)
return r
def is_gen(self):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: x.is_gen()
True
sage: (x+1).is_gen()
False
"""
cdef celement *gen = celement_new((<Polynomial_template>self)._cparent)
celement_gen(gen, 0, (<Polynomial_template>self)._cparent)
cdef bint r = celement_equal(&self.x, gen, (<Polynomial_template>self)._cparent)
celement_delete(gen, (<Polynomial_template>self)._cparent)
return r
def shift(self, int n):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: f = x^3 + x^2 + 1
sage: f.shift(1)
x^4 + x^3 + x
sage: f.shift(-1)
x^2 + x
"""
return element_shift(self, n)
def __lshift__(self, int n):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: f = x^3 + x^2 + 1
sage: f << 1
x^4 + x^3 + x
sage: f << -1
x^2 + x
"""
return element_shift(self, n)
def __rshift__(self, int n):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: x>>1
1
sage: (x^2 + x)>>1
x + 1
sage: (x^2 + x) >> -1
x^3 + x^2
"""
return element_shift(self, -n)
cpdef bint is_zero(self):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: x.is_zero()
False
"""
return celement_is_zero(&self.x, (<Polynomial_template>self)._cparent)
cpdef bint is_one(self):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: P(1).is_one()
True
"""
return celement_is_one(&self.x, (<Polynomial_template>self)._cparent)
def degree(self):
"""
EXAMPLES::
sage: P.<x> = GF(2)[]
sage: x.degree()
1
sage: P(1).degree()
0
sage: P(0).degree()
-1
"""
return Integer(celement_len(&self.x, (<Polynomial_template>self)._cparent)-1)
cpdef Polynomial truncate(self, long n):
r"""
Returns this polynomial mod `x^n`.
EXAMPLES::
sage: R.<x> =GF(2)[]
sage: f = sum(x^n for n in range(10)); f
x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
sage: f.truncate(6)
x^5 + x^4 + x^3 + x^2 + x + 1
If the precision is higher than the degree of the polynomial then
the polynomial itself is returned::
sage: f.truncate(10) is f
True
"""
if n >= celement_len(&self.x, (<Polynomial_template>self)._cparent):
return self
cdef type T = type(self)
cdef Polynomial_template r = <Polynomial_template>T.__new__(T)
celement_construct(&r.x, (<Polynomial_template>self)._cparent)
r._parent = (<Polynomial_template>self)._parent
r._cparent = (<Polynomial_template>self)._cparent
if n <= 0:
return r
cdef celement *gen = celement_new((<Polynomial_template>self)._cparent)
celement_gen(gen, 0, (<Polynomial_template>self)._cparent)
celement_pow(gen, gen, n, NULL, (<Polynomial_template>self)._cparent)
celement_mod(&r.x, &self.x, gen, (<Polynomial_template>self)._cparent)
celement_delete(gen, (<Polynomial_template>self)._cparent)
return r
def _singular_(self, singular=singular_default, have_ring=False):
r"""
Return Singular representation of this polynomial
INPUT:
- ``singular`` -- Singular interpreter (default: default interpreter)
- ``have_ring`` -- set to True if the ring was already set in Singular
EXAMPLES::
sage: P.<x> = PolynomialRing(GF(7))
sage: f = 3*x^2 + 2*x + 5
sage: singular(f)
3*x^2+2*x-2
"""
if not have_ring:
self.parent()._singular_(singular).set_ring() #this is expensive
return singular(self._singular_init_())
def _derivative(self, var=None):
r"""
Returns the formal derivative of self with respect to var.
var must be either the generator of the polynomial ring to which
this polynomial belongs, or None (either way the behaviour is the
same).
.. SEEALSO:: :meth:`.derivative`
EXAMPLES::
sage: R.<x> = Integers(77)[]
sage: f = x^4 - x - 1
sage: f._derivative()
4*x^3 + 76
sage: f._derivative(None)
4*x^3 + 76
sage: f._derivative(2*x)
Traceback (most recent call last):
...
ValueError: cannot differentiate with respect to 2*x
sage: y = var("y")
sage: f._derivative(y)
Traceback (most recent call last):
...
ValueError: cannot differentiate with respect to y
"""
if var is not None and var is not self._parent.gen():
raise ValueError("cannot differentiate with respect to %s" % var)
P = self.parent()
x = P.gen()
res = P(0)
for i,c in enumerate(self.list()[1:]):
res += (i+1)*c*x**i
return res
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