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

A dense polynomial over the integers, implemented via NTL.

__floordiv__()

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: f = R([9,6,1]) ; f
x^2 + 6*x + 9
sage: f // x
x + 6
sage: f // 3
2*x + 3
sage: g = x^3 ; g
x^3
sage: f // g
0
sage: g // f
x - 6
__getitem__()

Returns coefficient of x^n, or zero if n is negative.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: f = 2*x^2 - 3
sage: f[0]
-3
sage: f[1]
0
sage: f[2]
2
sage: f[3]
0
sage: f[-1]
0
__getslice__()

x.__getslice__(i, j) <==> x[i:j]

Use of negative indices is not supported.

__init__()
x.__init__(...) initializes x; see x.__class__.__doc__ for signature
static __new__()
T.__new__(S, ...) -> a new object with type S, a subtype of T
__reduce__()

Used for pickling.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: loads(dumps(x)) == x
True
sage: f = 2*x + 3
sage: loads(dumps(f)) == f
True
__rfloordiv__()
x.__rfloordiv__(y) <==> y//x
_add_()

Returns self plus right.

EXAMPLES:

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

Use NTL to factor self.

AUTHOR:

  • Joel B. Mohler

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: f = R([9,6,1])
sage: f._factor_ntl()
(x + 3)^2
_factor_pari()

Use pari to factor self.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: f = R([9,6,1]) ; f
x^2 + 6*x + 9
sage: f.factor()
(x + 3)^2
sage: f._factor_pari()
(x + 3)^2
_latex_()

Return the latex representation of this polynomial.

EXAMPLES:

sage: R.<t> = ZZ['t']
sage: latex(t^10-t^2-5*t+1)
t^{10} - t^{2} - 5t + 1
sage: latex(cyclotomic_polynomial(10^5))
x^{40000} - x^{30000} + x^{20000} - x^{10000} + 1
_lmul_()

Returns self multiplied by right, where right is a scalar (integer).

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: x*3
3*x
sage: (2*x^2 + 4)*3
6*x^2 + 12
_mul_()

Returns self multiplied by right.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: (x - 2)*(x^2 - 8*x + 16)
x^3 - 10*x^2 + 32*x - 32
_neg_()

Returns negative of self.

EXAMPLES:

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

EXAMPLES:

sage: t = PolynomialRing(ZZ,"t",implementation='NTL').gen()
sage: f = t^3 + 3*t - 17
sage: pari(f)
t^3 + 3*t - 17
_repr()

Return string representation of this polynomial.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, 'x', implementation='NTL')
sage: (-x+1)^5
-x^5 + 5*x^4 - 10*x^3 + 10*x^2 - 5*x + 1
_rmul_()

Returns self multiplied by right, where right is a scalar (integer).

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: 3*x
3*x
sage: 3*(2*x^2 + 4)
6*x^2 + 12
_sub_()

Return self minus right.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: f = 2*x + 1
sage: g = -3*x^2 + 6
sage: f - g
3*x^2 + 2*x - 5
_unsafe_mutate()

Sets coefficient of x^n to value.

This is very unsafe, because Sage polynomials are supposed to be immutable. (Shhhh don’t tell anyone!)

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ, implementation='NTL')
sage: f = 2*x^2 + 3; f
2*x^2 + 3
sage: f._unsafe_mutate(1, 42); f
2*x^2 + 42*x + 3
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()

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()

Return the discriminant of self, which is by definition

(-1)^{m(m-1)/2} {\mbox{\tt 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()

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()

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()

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()

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()

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()

Attempts 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)
...
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)
...
ArithmeticError: division not exact in Z[x] (consider coercing to Q[x] first)

TESTS:

sage: z = R(0)
sage: z.quo_rem(1)
(0, 0)
sage: z.quo_rem(x)
(0, 0)
sage: z.quo_rem(2*x)
(0, 0)
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()

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()

Return g, u, v such that g = u*self + v*right.

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

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