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)
<class 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>
sage: R.<a> = PolynomialRing(Integers(5*2^64))
sage: type(a)
<class '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

>>> from sage.all import *
>>> R = PolynomialRing(Integers(Integer(100)), names=('a',)); (a,) = R._first_ngens(1)
>>> type(a)
<class 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>
>>> R = PolynomialRing(Integers(Integer(5)*Integer(2)**Integer(64)), names=('a',)); (a,) = R._first_ngens(1)
>>> type(a)
<class 'sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_modn_ntl_ZZ'>
>>> R = PolynomialRing(Integers(Integer(5)*Integer(2)**Integer(64)), implementation="FLINT", names=('a',)); (a,) = R._first_ngens(1)
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[source]#

Bases: 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 get_unsafe(). See Polynomial_GF2X for an example.

degree()[source]#

EXAMPLES:

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

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> x.degree()
1
>>> P(Integer(1)).degree()
0
>>> P(Integer(0)).degree()
-1
gcd(other)[source]#

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

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> f = x*(x+Integer(1))
>>> f.gcd(x+Integer(1))
x + 1
>>> f.gcd(x**Integer(2))
x
get_cparent()[source]#
is_gen()[source]#

EXAMPLES:

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

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> x.is_gen()
True
>>> (x+Integer(1)).is_gen()
False
is_one()[source]#

EXAMPLES:

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

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> P(Integer(1)).is_one()
True
is_zero()[source]#

EXAMPLES:

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

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> x.is_zero()
False
list(copy=True)[source]#

EXAMPLES:

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

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> x.list()
[0, 1]
>>> list(x)
[0, 1]
quo_rem(right)[source]#

EXAMPLES:

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

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> f = x**Integer(2) + x + Integer(1)
>>> f.quo_rem(x + Integer(1))
(x, 1)
shift(n)[source]#

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

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> f = x**Integer(3) + x**Integer(2) + Integer(1)
>>> f.shift(Integer(1))
x^4 + x^3 + x
>>> f.shift(-Integer(1))
x^2 + x
truncate(n)[source]#

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

>>> from sage.all import *
>>> R = GF(Integer(2))['x']; (x,) = R._first_ngens(1)
>>> f = sum(x**n for n in range(Integer(10))); f
x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
>>> f.truncate(Integer(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

>>> from sage.all import *
>>> f.truncate(Integer(10)) is f
True

If the precision is negative, the zero polynomial is returned:

sage: f.truncate(-1)
0

>>> from sage.all import *
>>> f.truncate(-Integer(1))
0
xgcd(other)[source]#

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)

>>> from sage.all import *
>>> P = GF(Integer(7))['x']; (x,) = P._first_ngens(1)
>>> f = x*(x+Integer(1))
>>> f.xgcd(x+Integer(1))
(x + 1, 0, 1)
>>> f.xgcd(x**Integer(2))
(x, 1, 6)
class sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint[source]#

Bases: Polynomial_template

Polynomial on \(\ZZ/n\ZZ\) implemented via FLINT.

_add_(right)[source]#

EXAMPLES:

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

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> x + Integer(1)
x + 1
_sub_(right)[source]#

EXAMPLES:

sage: P.<x> = GF(2)[]
sage: x - 1
x + 1

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> x - Integer(1)
x + 1
_lmul_(left)[source]#

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

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> t = x**Integer(2) + x + Integer(1)
>>> Integer(0)*t
0
>>> Integer(1)*t
x^2 + x + 1

>>> R = GF(Integer(5))['y']; (y,) = R._first_ngens(1)
>>> u = y**Integer(2) + y + Integer(1)
>>> Integer(3)*u
3*y^2 + 3*y + 3
>>> Integer(5)*u
0
>>> (Integer(2)**Integer(81))*u
2*y^2 + 2*y + 2
>>> (-Integer(2)**Integer(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

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> t = x**Integer(2) + x + Integer(1)
>>> t*Integer(0)
0
>>> t*Integer(1)
x^2 + x + 1

>>> R = GF(Integer(5))['y']; (y,) = R._first_ngens(1)
>>> u = y**Integer(2) + y + Integer(1)
>>> u*Integer(3)
3*y^2 + 3*y + 3
>>> u*Integer(5)
0
_rmul_(right)[source]#

Multiply self on the right by a scalar.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: f = (x^3 + x + 5)
sage: f._rmul_(7)
7*x^3 + 7*x + 35
sage: f*7
7*x^3 + 7*x + 35

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> f = (x**Integer(3) + x + Integer(5))
>>> f._rmul_(Integer(7))
7*x^3 + 7*x + 35
>>> f*Integer(7)
7*x^3 + 7*x + 35
_mul_(right)[source]#

EXAMPLES:

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

>>> from sage.all import *
>>> P = GF(Integer(2))['x']; (x,) = P._first_ngens(1)
>>> x*(x+Integer(1))
x^2 + x
_mul_trunc_(right, n)[source]#

Return the product of this polynomial and other truncated to the given length \(n\).

This function is usually more efficient than simply doing the multiplication and then truncating. The function is tuned for length \(n\) about half the length of a full product.

EXAMPLES:

sage: P.<a> = GF(7)[]
sage: a = P(range(10)); b = P(range(5, 15))
sage: a._mul_trunc_(b, 5)
4*a^4 + 6*a^3 + 2*a^2 + 5*a

>>> from sage.all import *
>>> P = GF(Integer(7))['a']; (a,) = P._first_ngens(1)
>>> a = P(range(Integer(10))); b = P(range(Integer(5), Integer(15)))
>>> a._mul_trunc_(b, Integer(5))
4*a^4 + 6*a^3 + 2*a^2 + 5*a
factor()[source]#

Return the factorization of the polynomial.

EXAMPLES:

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

>>> from sage.all import *
>>> R = GF(Integer(5))['x']; (x,) = R._first_ngens(1)
>>> (x**Integer(2) + Integer(1)).factor()
(x + 2) * (x + 3)

It also works for prime-power moduli:

sage: R.<x> = Zmod(23^5)[]
sage: (x^3 + 1).factor()
(x + 1) * (x^2 + 6436342*x + 1)

>>> from sage.all import *
>>> R = Zmod(Integer(23)**Integer(5))['x']; (x,) = R._first_ngens(1)
>>> (x**Integer(3) + Integer(1)).factor()
(x + 1) * (x^2 + 6436342*x + 1)
is_irreducible()[source]#

Return whether 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

>>> from sage.all import *
>>> R = GF(Integer(5))['x']; (x,) = R._first_ngens(1)
>>> (x**Integer(2) + Integer(1)).is_irreducible()
False
>>> (x**Integer(3) + x + Integer(1)).is_irreducible()
True

Not implemented when the base ring is not a field:

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

>>> from sage.all import *
>>> S = Zmod(Integer(10))['s']; (s,) = S._first_ngens(1)
>>> (s**Integer(2)).is_irreducible()
Traceback (most recent call last):
...
NotImplementedError: checking irreducibility of polynomials
over rings with composite characteristic is not implemented
minpoly_mod(other)[source]#

Thin wrapper for sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_n.minpoly_mod().

EXAMPLES:

sage: R.<x> = GF(127)[]
sage: type(x)
<class 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>
sage: (x^5 - 3).minpoly_mod(x^3 + 5*x - 1)
x^3 + 34*x^2 + 125*x + 95

>>> from sage.all import *
>>> R = GF(Integer(127))['x']; (x,) = R._first_ngens(1)
>>> type(x)
<class 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>
>>> (x**Integer(5) - Integer(3)).minpoly_mod(x**Integer(3) + Integer(5)*x - Integer(1))
x^3 + 34*x^2 + 125*x + 95
monic()[source]#

Return this polynomial divided by its leading coefficient.

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

EXAMPLES:

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

>>> from sage.all import *
>>> R = GF(Integer(5))['x']; (x,) = R._first_ngens(1)
>>> (Integer(2)*x**Integer(2) + Integer(1)).monic()
x^2 + 3
rational_reconstruct(*args, **kwds)[source]#

Deprecated: Use rational_reconstruction() instead. See Issue #12696 for details.

rational_reconstruction(m, n_deg=0, d_deg=0)[source]#

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_reconstruction(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

>>> from sage.all import *
>>> P = GF(Integer(5))['x']; (x,) = P._first_ngens(1)
>>> p = Integer(4)*x**Integer(5) + Integer(3)*x**Integer(4) + Integer(2)*x**Integer(3) + Integer(2)*x**Integer(2) + Integer(4)*x + Integer(2)
>>> n, d = p.rational_reconstruction(x**Integer(9), Integer(4), Integer(4)); n, d
(3*x^4 + 2*x^3 + x^2 + 2*x, x^4 + 3*x^3 + x^2 + x)
>>> (p*d % x**Integer(9)) == n
True

Check that Issue #37169 is fixed - it does not throw an error:

sage: R.<x> = Zmod(4)[]
sage: _.<z> = R.quotient_ring(x^2 - 1)
sage: c = 2 * z + 1
sage: c * Zmod(2).zero()
Traceback (most recent call last):
...
RuntimeError: Aborted

>>> from sage.all import *
>>> R = Zmod(Integer(4))['x']; (x,) = R._first_ngens(1)
>>> _ = R.quotient_ring(x**Integer(2) - Integer(1), names=('z',)); (z,) = _._first_ngens(1)
>>> c = Integer(2) * z + Integer(1)
>>> c * Zmod(Integer(2)).zero()
Traceback (most recent call last):
...
RuntimeError: Aborted
resultant(other)[source]#

Return 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

>>> from sage.all import *
>>> R = GF(Integer(19))['x']; (x,) = R._first_ngens(1)
>>> f = x**Integer(3) + x + Integer(1);  g = x**Integer(3) - x - Integer(1)
>>> r = f.resultant(g); r
11
>>> r.parent() is GF(Integer(19))
True

The following example shows that Issue #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

>>> from sage.all import *
>>> R = ZZ.quo(Integer(9))['x']; (x,) = R._first_ngens(1)
>>> f = Integer(2)*x**Integer(3) + x**Integer(2) + x;  g = Integer(6)*x**Integer(2) + Integer(2)*x + Integer(1)
>>> f.resultant(g)
5
reverse(degree=None)[source]#

Return a polynomial with the coefficients of this polynomial reversed.

If the optional argument degree is given, the coefficient list will be truncated or zero padded as necessary before computing the reverse.

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

>>> from sage.all import *
>>> R = GF(Integer(5))['x']; (x,) = R._first_ngens(1)
>>> p = R([Integer(1),Integer(2),Integer(3),Integer(4)]); p
4*x^3 + 3*x^2 + 2*x + 1
>>> p.reverse()
x^3 + 2*x^2 + 3*x + 4
>>> p.reverse(degree=Integer(6))
x^6 + 2*x^5 + 3*x^4 + 4*x^3
>>> p.reverse(degree=Integer(2))
x^2 + 2*x + 3

>>> R = GF(Integer(101))['x']; (x,) = R._first_ngens(1)
>>> f = x**Integer(3) - x + Integer(2); f
x^3 + 100*x + 2
>>> f.reverse()
2*x^3 + 100*x^2 + 1
>>> f.reverse() == f(Integer(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

>>> from sage.all import *
>>> f = x**Integer(3) + Integer(2)*x
>>> 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

>>> from sage.all import *
>>> f
x^3 + 2*x
>>> f.reverse().reverse()
x^2 + 2
>>> f.reverse(Integer(5)).reverse(Integer(5))
x^3 + 2*x
revert_series(n)[source]#

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

>>> from sage.all import *
>>> R = GF(Integer(5))['t']; (t,) = R._first_ngens(1)
>>> f = t + Integer(2)*t**Integer(2) - t**Integer(3) - Integer(3)*t**Integer(4)
>>> f.revert_series(Integer(5))
3*t^4 + 4*t^3 + 3*t^2 + t

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

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

>>> g = t + Integer(2)*t**Integer(2) - t**Integer(3) -Integer(3)*t**Integer(4) + t**Integer(5)
>>> g.revert_series(Integer(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)[source]#

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

EXAMPLES:

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]

>>> from sage.all import *
>>> N = Integer(10001)
>>> K = Zmod(Integer(10001))
>>> P = PolynomialRing(K, names=('x',)); (x,) = P._first_ngens(1)
>>> f = x**Integer(3) + Integer(10)*x**Integer(2) + Integer(5000)*x - Integer(222)
>>> f.small_roots()
[4]
squarefree_decomposition()[source]#

Return 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

>>> from sage.all import *
>>> R = GF(Integer(5))['x']; (x,) = R._first_ngens(1)
>>> ((x+Integer(1))*(x**Integer(2)+Integer(1))**Integer(2)*x**Integer(3)).squarefree_decomposition()
(x + 1) * (x^2 + 1)^2 * x^3
sage.rings.polynomial.polynomial_zmod_flint.make_element(parent, args)[source]#