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

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 over the completion at \(p\).

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))

A more difficult example:

sage: f = 100 * (5*x + 1)^2 * (x + 5)^2
sage: f.factor_padic(5, 10)
(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()                                               # needs sage.libs.linbox
[((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