# Dense univariate polynomials over $$\ZZ$$, 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 $$\ZZ$$; 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: 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

$(-1)^{m(m-1)/2} \text{resultant}(a, a')/lc(a),$

where $$m = deg(a)$$, and $$lc(a)$$ 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 $$2^{-80}$$.

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)
Traceback (most recent call last):
...
ArithmeticError: factorization of 0 is not defined

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


Return $$p$$-adic factorization of self to given precision.

INPUT:

• p – prime

• prec – integer; the precision

OUTPUT:

• factorization of self over the completion at $$p$$.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: f = x^2 + 1
((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))


A more difficult example:

sage: f = 100 * (5*x + 1)^2 * (x + 5)^2
(4 + O(5^10)) * (5 + O(5^11))^2 * ((1 + O(5^10))*x + 5 + O(5^10))^2
* ((5 + O(5^10))*x + 1 + O(5^10))^2

gcd(right)#

Return the GCD of self and right. The leading coefficient need not be 1.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: f = (6*x + 47) * (7*x^2 - 2*x + 38)
sage: g = (6*x + 47) * (3*x^3 + 2*x + 1)
sage: f.gcd(g)
6*x + 47

lcm(right)#

Return the LCM of self and right.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: f = (6*x + 47) * (7*x^2 - 2*x + 38)
sage: g = (6*x + 47) * (3*x^3 + 2*x + 1)
sage: h = f.lcm(g); h
126*x^6 + 951*x^5 + 486*x^4 + 6034*x^3 + 585*x^2 + 3706*x + 1786
sage: h == (6*x + 47) * (7*x^2 - 2*x + 38) * (3*x^3 + 2*x + 1)
True

list(copy=True)#

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)#

Attempt to divide self by right, and return a quotient and remainder.

If right is monic, then it returns (q, r) where self = q * right + r and deg(r) < deg(right).

If right is not monic, then it returns $$(q, 0)$$ where q = self/right if right exactly divides self, otherwise it raises an exception.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: f = R(range(10)); g = R([-1, 0, 1])
sage: q, r = f.quo_rem(g)
sage: q, r
(9*x^7 + 8*x^6 + 16*x^5 + 14*x^4 + 21*x^3 + 18*x^2 + 24*x + 20, 25*x + 20)
sage: q*g + r == f
True

sage: 0//(2*x)
0

sage: f = x^2
sage: f.quo_rem(0)
Traceback (most recent call last):
...
ArithmeticError: division by zero polynomial

sage: f = (x^2 + 3) * (2*x - 1)
sage: f.quo_rem(2*x - 1)
(x^2 + 3, 0)

sage: f = x^2
sage: f.quo_rem(2*x - 1)
Traceback (most recent call last):
...
ArithmeticError: division not exact in Z[x] (consider coercing to Q[x] first)

real_root_intervals()#

Returns isolating intervals for the real roots of this polynomial.

EXAMPLES: 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)#

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

If proof=False (the default is proof=True), then this function may use a randomized strategy that errors with probability no more than $$2^{-80}$$.

INPUT:

• other – a polynomial

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

EXAMPLES:

sage: x = PolynomialRing(ZZ, 'x', implementation='NTL').0
sage: f = x^3 + x + 1;  g = x^3 - x - 1
sage: r = f.resultant(g); r
-8
sage: r.parent() is ZZ
True

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)#

This function can’t in general return $$(g,s,t)$$ as above, since they need not exist. Instead, over the integers, we first multiply $$g$$ by a divisor of the resultant of $$a/g$$ and $$b/g$$, up to sign, and return g, u, v such that g = s*self + s*right. But note that this $$g$$ may be a multiple of the gcd.

If self and right are coprime as polynomials over the rationals, then g is guaranteed to be the resultant of self and right, as a constant polynomial.

EXAMPLES:

sage: P.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: F = (x^2 + 2)*x^3; G = (x^2+2)*(x-3)
sage: g, u, v = F.xgcd(G)
sage: g, u, v
(27*x^2 + 54, 1, -x^2 - 3*x - 9)
sage: u*F + v*G
27*x^2 + 54
sage: x.xgcd(P(0))
(x, 1, 0)
sage: f = P(0)
sage: f.xgcd(x)
(x, 0, 1)
sage: F = (x-3)^3; G = (x-15)^2
sage: g, u, v = F.xgcd(G)
sage: g, u, v
(2985984, -432*x + 8208, 432*x^2 + 864*x + 14256)
sage: u*F + v*G
2985984