Dense univariate polynomials over \(\ZZ/n\ZZ\), implemented using FLINT.

Dense univariate polynomials over \(\ZZ/n\ZZ\), implemented using FLINT.

This module gives a fast implementation of \((\ZZ/n\ZZ)[x]\) whenever \(n\) is at most sys.maxsize. We use it by default in preference to NTL when the modulus is small, falling back to NTL if the modulus is too large, as in the example below.

EXAMPLES:

sage: R.<a> = PolynomialRing(Integers(100))
sage: type(a)
<type 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>
sage: R.<a> = PolynomialRing(Integers(5*2^64))
sage: type(a)
<type 'sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_modn_ntl_ZZ'>
sage: R.<a> = PolynomialRing(Integers(5*2^64), implementation="FLINT")
Traceback (most recent call last):
...
ValueError: FLINT does not support modulus 92233720368547758080

AUTHORS:

  • Burcin Erocal (2008-11) initial implementation
  • Martin Albrecht (2009-01) another initial implementation
class sage.rings.polynomial.polynomial_zmod_flint.Polynomial_template

Bases: sage.rings.polynomial.polynomial_element.Polynomial

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 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 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 __getitem__. See Polynomial_GF2X for an example.

degree()

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: x.degree()
1
sage: P(1).degree()
0
sage: P(0).degree()
-1
gcd(other)

Return the greatest common divisor of self and other.

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: f = x*(x+1)
sage: f.gcd(x+1)
x + 1
sage: f.gcd(x^2)
x
get_cparent()
is_gen()

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: x.is_gen()
True
sage: (x+1).is_gen()
False
is_one()

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: P(1).is_one()
True
is_zero()

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: x.is_zero()
False
list()

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: x.list()
[0, 1]
sage: list(x)
[0, 1]
quo_rem(right)

EXAMPLE:

sage: P.<x> = GF(2)[]
sage: f = x^2 + x + 1
sage: f.quo_rem(x + 1)
(x, 1)
shift(n)

EXAMPLE:

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
truncate(n)

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
xgcd(other)

Computes extended gcd of self and other.

EXAMPLE:

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)
class sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint

Bases: sage.rings.polynomial.polynomial_zmod_flint.Polynomial_template

EXAMPLE:

sage: P.<x> = GF(32003)[]
sage: f = 24998*x^2 + 29761*x + 2252
factor()

Returns the factorization of the polynomial.

EXAMPLES:

sage: R.<x> = GF(5)[]
sage: (x^2 + 1).factor()
(x + 2) * (x + 3)

TESTS:

sage: (2*x^2 + 2).factor()
(2) * (x + 2) * (x + 3)
sage: P.<x> = Zmod(10)[]
sage: (x^2).factor()
Traceback (most recent call last):
...
NotImplementedError: factorization of polynomials over rings with composite characteristic is not implemented
is_irreducible()

Return True if this polynomial is irreducible.

EXAMPLES:

sage: R.<x> = GF(5)[]
sage: (x^2 + 1).is_irreducible()
False
sage: (x^3 + x + 1).is_irreducible()
True

TESTS:

sage: R(0).is_irreducible()
False
sage: R(1).is_irreducible()
False
sage: R(2).is_irreducible()
False

sage: S.<s> = Zmod(10)[]
sage: (s^2).is_irreducible()
Traceback (most recent call last):
...
NotImplementedError: checking irreducibility of polynomials over rings with composite characteristic is not implemented
sage: S(1).is_irreducible()
False
sage: S(2).is_irreducible()
Traceback (most recent call last):
...
NotImplementedError: checking irreducibility of polynomials over rings with composite characteristic is not implemented
monic()

Return this polynomial divided by its leading coefficient.

Raises ValueError if the leading cofficient is not invertible in the base ring.

EXAMPLES:

sage: R.<x> = GF(5)[]
sage: (2*x^2+1).monic()
x^2 + 3

TESTS:

sage: R.<x> = Zmod(10)[]
sage: (5*x).monic()
Traceback (most recent call last):
...
ValueError: leading coefficient must be invertible
rational_reconstruct(m, n_deg=0, d_deg=0)

Construct a rational function n/d such that \(p*d\) is equivalent to \(n\) modulo \(m\) where \(p\) is this polynomial.

EXAMPLES:

sage: P.<x> = GF(5)[]
sage: p = 4*x^5 + 3*x^4 + 2*x^3 + 2*x^2 + 4*x + 2
sage: n, d = p.rational_reconstruct(x^9, 4, 4); n, d
(3*x^4 + 2*x^3 + x^2 + 2*x, x^4 + 3*x^3 + x^2 + x)
sage: (p*d % x^9) == n
True
resultant(other)

Returns the resultant of self and other, which must lie in the same polynomial ring.

INPUT:

  • other – a polynomial

OUTPUT: an element of the base ring of the polynomial ring

EXAMPLES:

sage: R.<x> = GF(19)['x']
sage: f = x^3 + x + 1;  g = x^3 - x - 1
sage: r = f.resultant(g); r
11
sage: r.parent() is GF(19)
True

The following example shows that #11782 has been fixed:

sage: R.<x> = ZZ.quo(9)['x']
sage: f = 2*x^3 + x^2 + x;  g = 6*x^2 + 2*x + 1
sage: f.resultant(g)
5
reverse(degree=None)

Return a polynomial with the coefficients of this polynomial reversed.

If an optional degree argument is given the coefficient list will be truncated or zero padded as necessary and the reverse polynomial will have the specified degree.

EXAMPLES:

sage: R.<x> = GF(5)[]
sage: p = R([1,2,3,4]); p
4*x^3 + 3*x^2 + 2*x + 1
sage: p.reverse()
x^3 + 2*x^2 + 3*x + 4
sage: p.reverse(degree=6)
x^6 + 2*x^5 + 3*x^4 + 4*x^3
sage: p.reverse(degree=2)
x^2 + 2*x + 3

sage: R.<x> = GF(101)[]
sage: f = x^3 - x + 2; f
x^3 + 100*x + 2
sage: f.reverse()
2*x^3 + 100*x^2 + 1
sage: f.reverse() == f(1/x) * x^f.degree()
True

Note that if \(f\) has zero constant coefficient, its reverse will have lower degree.

sage: f = x^3 + 2*x
sage: f.reverse()
2*x^2 + 1

In this case, reverse is not an involution unless we explicitly specify a degree.

sage: f
x^3 + 2*x
sage: f.reverse().reverse()
x^2 + 2
sage: f.reverse(5).reverse(5)
x^3 + 2*x

TESTS:

sage: p.reverse(degree=1.5r)
Traceback (most recent call last):
...
ValueError: degree argument must be a non-negative integer, got 1.5
revert_series(n)

Return a polynomial \(f\) such that \(f(self(x)) = self(f(x)) = x mod x^n\).

EXAMPLES:

sage: R.<t> =  GF(5)[]
sage: f = t + 2*t^2 - t^3 - 3*t^4
sage: f.revert_series(5)
3*t^4 + 4*t^3 + 3*t^2 + t

sage: f.revert_series(-1)
Traceback (most recent call last):
...
ValueError: argument n must be a non-negative integer, got -1

sage: g = - t^3 + t^5
sage: g.revert_series(6)
Traceback (most recent call last):
...
ValueError: self must have constant coefficient 0 and a unit for coefficient t^1

sage: g = t + 2*t^2 - t^3 -3*t^4 + t^5
sage: g.revert_series(6)
Traceback (most recent call last):
...
ValueError: the integers 1 up to n=5 are required to be invertible over the base field
small_roots(*args, **kwds)

See sage.rings.polynomial.polynomial_modn_dense_ntl.small_roots() for the documentation of this function.

EXAMPLE:

sage: N = 10001
sage: K = Zmod(10001)
sage: P.<x> = PolynomialRing(K)
sage: f = x^3 + 10*x^2 + 5000*x - 222
sage: f.small_roots()
[4]
squarefree_decomposition()

Returns the squarefree decomposition of this polynomial.

EXAMPLES:

sage: R.<x> = GF(5)[]
sage: ((x+1)*(x^2+1)^2*x^3).squarefree_decomposition()
(x + 1) * (x^2 + 1)^2 * x^3

TESTS:

sage: (2*x*(x+1)^2).squarefree_decomposition()
(2) * x * (x + 1)^2
sage: P.<x> = Zmod(10)[]
sage: (x^2).squarefree_decomposition()
Traceback (most recent call last):
...
NotImplementedError: square free factorization of polynomials over rings with composite characteristic is not implemented
sage.rings.polynomial.polynomial_zmod_flint.make_element(parent, args)

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