This module gives a fast implementation of whenever
is at
most sys.maxint. 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")
...
ValueError: FLINT does not support modulus 92233720368547758080
AUTHORS:
Template for interfacing to external C / C++ libraries for implementations of polynomials.
AUTHORS:
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
.
Note
Implementations using this template MUST implement coercion from base ring elements and __getitem__. See Polynomial_GF2X for an example.
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: copy(x) is x
False
sage: copy(x) == x
True
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: x//(x + 1)
1
sage: (x + 1)//x
1
x.__getslice__(i, j) <==> x[i:j]
Use of negative indices is not supported.
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: int(x)
...
ValueError: Cannot coerce polynomial with degree 1 to integer.
sage: int(P(1))
1
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: f = x^3 + x^2 + 1
sage: f << 1
x^4 + x^3 + x
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: (x^2 + 1) % x^2
1
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: -x
x
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: loads(dumps(x)) == x
True
EXAMPLE:
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
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: x + 1
x + 1
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).
See also
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)
...
ValueError: cannot differentiate with respect to 2*x
sage: y = var("y")
sage: f._derivative(y)
...
ValueError: cannot differentiate with respect to y
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: x*(x+1)
x^2 + x
Return Singular representation of this polynomial
INPUT:
EXAMPLE:
sage: P.<x> = PolynomialRing(GF(7))
sage: f = 3*x^2 + 2*x + 5
sage: singular(f)
3*x^2+2*x-2
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: x - 1
x + 1
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: x.degree()
1
sage: P(1).degree()
0
sage: P(0).degree()
-1
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
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: x.is_gen()
True
sage: (x+1).is_gen()
False
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: P(1).is_one()
True
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: x.is_zero()
False
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: x.list()
[0, 1]
sage: list(x)
[0, 1]
EXAMPLE:
sage: P.<x> = GF(2)[]
sage: f = x^2 + x + 1
sage: f.quo_rem(x + 1)
(x, 1)
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
Returns this polynomial mod .
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
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)
Evaluate polynomial at x=a.
INPUT: either
EXAMPLE:
sage: P.<x> = PolynomialRing(GF(7))
sage: f= x^2 + 1
sage: f(0)
1
sage: f(2)
5
sage: f(3)
3
EXAMPLE:
sage: P.<x> = GF(32003)[]
sage: f = 24998*x^2 + 29761*x + 2252
sage: f[100]
0
sage: f[1]
29761
sage: f[0]
2252
sage: f[-1]
0
Returns the product of two polynomials using the zn_poly library.
See http://www.math.harvard.edu/~dmharvey/zn_poly/ for details on zn_poly.
INPUT:
OUTPUT: (Polynomial) the product self*right.
EXAMPLE:
sage: P.<x> = PolynomialRing(GF(next_prime(2^30)))
sage: f = P.random_element(1000)
sage: g = P.random_element(1000)
sage: f*g == f._mul_zn_poly(g)
True
sage: P.<x> = PolynomialRing(Integers(100))
sage: P
Univariate Polynomial Ring in x over Ring of integers modulo 100
sage: r = (10*x)._mul_zn_poly(10*x); r
0
sage: r.degree()
-1
ALGORITHM:
uses David Harvey’s zn_poly library.
NOTE: This function is a technology preview. It might disappear or be replaced without a deprecation warning.
Never use this unless you really know what you are doing.
INPUT:
Warning
This could easily introduce subtle bugs, since Sage assumes everywhere that polynomials are immutable. It’s OK to use this if you really know what you’re doing.
EXAMPLES:
sage: R.<x> = GF(7)[]
sage: f = (1+2*x)^2; f
4*x^2 + 4*x + 1
sage: f._unsafe_mutate(1, -5)
sage: f
4*x^2 + 2*x + 1
Returns the resultant of self and other, which must lie in the same polynomial ring.
INPUT:
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
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]