Dense univariate polynomials over \(\ZZ\), implemented using FLINT.

Dense univariate polynomials over \(\ZZ\), implemented using FLINT.

AUTHORS:

  • 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
  • David Harvey: split off from polynomial_element_generic.py (2007-09)
  • Burcin Erocal: rewrote to use FLINT (2008-06-16)
class sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_flint

Bases: sage.rings.polynomial.polynomial_element.Polynomial

A dense polynomial over the integers, implemented via FLINT.

content()

Return the greatest common divisor of the coefficients of this polynomial.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ)
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

TESTS:

sage: t = x^2+x+1
sage: t.content()
1
sage: (123456789123456789123456789123456789123456789*t).content()
123456789123456789123456789123456789123456789
degree(gen=None)

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

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ)
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

\[(-1)^{m(m-1)/2} \mathop{\mathrm{resultant}}(a, a')/\mathop{\mathrm{lc}}(a),\]

where \(m = \mathop{\mathrm{deg}}(a)\), and \(\mathop{\mathrm{lc}}(a)\) 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: R.<x> = ZZ[]
sage: f = 3*x^3 + 2*x + 1
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> = ZZ['x']
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):
...
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 over the completion at \(p\).

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ)
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)
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
is_zero()

Returns True if self is equal to zero.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: R(0).is_zero()
True
sage: R(1).is_zero()
False
sage: x.is_zero()
False
lcm(right)

Return the LCM of self and right.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ)
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').0
sage: f = x^3 + 3*x - 17
sage: f.list()
[-17, 3, 0, 1]
sage: f = PolynomialRing(ZZ,'x')(0)
sage: f.list()
[]
pseudo_divrem(B)

Write A = self. This function computes polynomials \(Q\) and \(R\) and an integer \(d\) such that

\[\mathop{\mathrm{lead}}(B)^d A = B Q + R\]

where R has degree less than that of B.

INPUT:

  • B – a polynomial over \(\ZZ\)

OUTPUT:

  • Q, R – polynomials
  • d – nonnegative integer

EXAMPLES:

sage: R.<x> = ZZ['x']
sage: A = R(range(10)); B = 3*R([-1, 0, 1])
sage: Q, R, d = A.pseudo_divrem(B)
sage: Q, R, d
(9*x^7 + 8*x^6 + 16*x^5 + 14*x^4 + 21*x^3 + 18*x^2 + 24*x + 20, 75*x + 60, 1)
sage: B.leading_coefficient()^d * A == B*Q + R
True
quo_rem(right)

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

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ)
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: f = x^2
sage: f.quo_rem(0)
Traceback (most recent call last):
...
ZeroDivisionError: 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)
(0, x^2)

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)

Ticket #383, make sure things get coerced correctly:

sage: f = x+1; parent(f)
Univariate Polynomial Ring in x over Integer Ring
sage: g = x/2; parent(g)
Univariate Polynomial Ring in x over Rational Field
sage: f.quo_rem(g)
(2, 1)
sage: g.quo_rem(f)
(1/2, -1/2)
sage: parent(f.quo_rem(g)[0])
Univariate Polynomial Ring in x over Rational Field
sage: f.quo_rem(3)
(0, x + 1)
sage: (5*x+7).quo_rem(3)
(x + 2, 2*x + 1)
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)
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').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
reverse(degree=None)

Return a polynomial with the coefficients of this polynomial reversed.

If an optional degree argument is given the coefficient list will be truncated or zero padded as necessary and the reverse polynomial will have the specified degree.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: p = R([1,2,3,4]); p
4*x^3 + 3*x^2 + 2*x + 1
sage: p.reverse()
x^3 + 2*x^2 + 3*x + 4
sage: p.reverse(degree=6)
x^6 + 2*x^5 + 3*x^4 + 4*x^3
sage: p.reverse(degree=2)
x^2 + 2*x + 3

TESTS:

sage: p.reverse(degree=1.5r)
Traceback (most recent call last):
...
ValueError: degree argument must be a non-negative integer, got 1.5
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)
sage: p = (x-1)^2 * (x-2)^2 * (x-3)^3 * (x-4)
sage: p.squarefree_decomposition()
(x - 4) * (x^2 - 3*x + 2)^2 * (x - 3)^3
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)
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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