Dense univariate polynomials over , implemented using NTL.

AUTHORS:

• David Harvey: split off from polynomial_element_generic.py (2007-09)
• David Harvey: rewrote to talk to NTL directly, instead of via ntl.pyx (2007-09); a lot of this was based on Joel Mohler’s recent rewrite of the NTL wrapper

Sage includes two implementations of dense univariate polynomials over ; this file contains the implementation based on NTL, but there is also an implementation based on FLINT in sage.rings.polynomial.polynomial_integer_dense_flint.

The FLINT implementation is preferred (FLINT’s arithmetic operations are generally faster), so it is the default; to use the NTL implementation, you can do:

sage: K.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: K
Univariate Polynomial Ring in x over Integer Ring (using NTL)

class sage.rings.polynomial.polynomial_integer_dense_ntl.Polynomial_integer_dense_ntl

Bases: sage.rings.polynomial.polynomial_element.Polynomial

A dense polynomial over the integers, implemented via NTL.

content()

Return the greatest common divisor of the coefficients of this polynomial. The sign is the sign of the leading coefficient. The content of the zero polynomial is zero.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: (2*x^2 - 4*x^4 + 14*x^7).content()
2
sage: (2*x^2 - 4*x^4 - 14*x^7).content()
-2
sage: x.content()
1
sage: R(1).content()
1
sage: R(0).content()
0

degree(gen=None)

Return the degree of this polynomial. The zero polynomial has degree -1.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: x.degree()
1
sage: (x^2).degree()
2
sage: R(1).degree()
0
sage: R(0).degree()
-1

discriminant(proof=True)

Return the discriminant of self, which is by definition

where , and is the leading coefficient of a. If proof is False (the default is True), then this function may use a randomized strategy that errors with probability no more than .

EXAMPLES:

sage: f = ntl.ZZX([1,2,0,3])
sage: f.discriminant()
-339
sage: f.discriminant(proof=False)
-339

factor()

This function overrides the generic polynomial factorization to make a somewhat intelligent decision to use Pari or NTL based on some benchmarking.

Note: This function factors the content of the polynomial, which can take very long if it’s a really big integer. If you do not need the content factored, divide it out of your polynomial before calling this function.

EXAMPLES:

sage: R.<x>=ZZ[]
sage: f=x^4-1
sage: f.factor()
(x - 1) * (x + 1) * (x^2 + 1)
sage: f=1-x
sage: f.factor()
(-1) * (x - 1)
sage: f.factor().unit()
-1
sage: f = -30*x; f.factor()
(-1) * 2 * 3 * 5 * x

factor_mod(p)

Return the factorization of self modulo the prime p.

INPUT:

• p – prime

OUTPUT: factorization of self reduced modulo p.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, 'x', implementation='NTL')
sage: f = -3*x*(x-2)*(x-9) + x
sage: f.factor_mod(3)
x
sage: f = -3*x*(x-2)*(x-9)
sage: f.factor_mod(3)
...
ValueError: factorization of 0 not defined

sage: f = 2*x*(x-2)*(x-9)
sage: f.factor_mod(7)
(2) * x * (x + 5)^2

factor_padic(p, prec=10)

Return p-adic factorization of self to given precision.

INPUT:

• p – prime
• prec – integer; the precision

OUTPUT: factorization of self reduced modulo p.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: f = x^2 + 1
sage: f.factor_padic(5, 4)
((1 + O(5^4))*x + (2 + 5 + 2*5^2 + 5^3 + O(5^4))) * ((1 + O(5^4))*x + (3 + 3*5 + 2*5^2 + 3*5^3 + O(5^4)))

gcd(right)

A decorator to be used on binary operation methods that should operate on elements of the same parent. If the parents of the arguments differ, coercion is performed, then the method is re-looked up by name on the first argument.

In short, using the NamedBinopMethod (alias coerce_binop) decorator on a method gives it the exact same semantics of the basic arithmetic operations like _add_, _sub_, etc. in that both operands are guaranteed to have exactly the same parent.

lcm(right)

A decorator to be used on binary operation methods that should operate on elements of the same parent. If the parents of the arguments differ, coercion is performed, then the method is re-looked up by name on the first argument.

In short, using the NamedBinopMethod (alias coerce_binop) decorator on a method gives it the exact same semantics of the basic arithmetic operations like _add_, _sub_, etc. in that both operands are guaranteed to have exactly the same parent.

list()

Return a new copy of the list of the underlying elements of self.

EXAMPLES:

sage: x = PolynomialRing(ZZ,'x',implementation='NTL').0
sage: f = x^3 + 3*x - 17
sage: f.list()
[-17, 3, 0, 1]
sage: f = PolynomialRing(ZZ,'x',implementation='NTL')(0)
sage: f.list()
[]

quo_rem(right)

A decorator to be used on binary operation methods that should operate on elements of the same parent. If the parents of the arguments differ, coercion is performed, then the method is re-looked up by name on the first argument.

In short, using the NamedBinopMethod (alias coerce_binop) decorator on a method gives it the exact same semantics of the basic arithmetic operations like _add_, _sub_, etc. in that both operands are guaranteed to have exactly the same parent.

real_root_intervals()

Returns isolating intervals for the real roots of this polynomial.

EXAMPLE: We compute the roots of the characteristic polynomial of some Salem numbers:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: f = 1 - x^2 - x^3 - x^4 + x^6
sage: f.real_root_intervals()
[((1/2, 3/4), 1), ((1, 3/2), 1)]

resultant(other, proof=True)

A decorator to be used on binary operation methods that should operate on elements of the same parent. If the parents of the arguments differ, coercion is performed, then the method is re-looked up by name on the first argument.

In short, using the NamedBinopMethod (alias coerce_binop) decorator on a method gives it the exact same semantics of the basic arithmetic operations like _add_, _sub_, etc. in that both operands are guaranteed to have exactly the same parent.

squarefree_decomposition()

Return the square-free decomposition of self. This is a partial factorization of self into square-free, relatively prime polynomials.

This is a wrapper for the NTL function SquareFreeDecomp.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: p = 37 * (x-1)^2 * (x-2)^2 * (x-3)^3 * (x-4)
sage: p.squarefree_decomposition()
(37) * (x - 4) * (x^2 - 3*x + 2)^2 * (x - 3)^3

xgcd(right)

A decorator to be used on binary operation methods that should operate on elements of the same parent. If the parents of the arguments differ, coercion is performed, then the method is re-looked up by name on the first argument.

In short, using the NamedBinopMethod (alias coerce_binop) decorator on a method gives it the exact same semantics of the basic arithmetic operations like _add_, _sub_, etc. in that both operands are guaranteed to have exactly the same parent.