"""
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
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) except -1:
        """
        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) except -1:
        """
        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_())
