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/*
Copyright (C) 2016 Vincent Delecroix
This file is part of FLINT.
FLINT is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "fmpz.h"
#include "fmpz_vec.h"
#include "fmpz_poly.h"
int _fmpz_poly_has_real_root(const fmpz * p, slong len)
{
slong n, i;
int s, t;
/* O(1) conditions: */
/* - constant polynomial */
/* - odd degree */
/* - p(0) = 0 */
/* - sign(p(0)) * sign(p(+infinity)) = -1 */
if (len == 1)
return 0;
if (len % 2 == 0)
return 1;
if (fmpz_is_zero(p) || (fmpz_sgn(p) * fmpz_sgn(p + len - 1) < 0))
return 1;
/* O(len) conditions: Descartes rule of sign */
n = 0;
s = fmpz_sgn(p);
for (i = 1; i < len; i++)
{
if (fmpz_is_zero(p + i)) continue;
t = fmpz_sgn(p + i);
if (t != s)
{
n += 1;
s = t;
}
}
if (n % 2 == 1) return 1;
n = 0;
s = fmpz_sgn(p);
for (i = 1; i < len; i++)
{
if (fmpz_is_zero(p + i)) continue;
t = fmpz_sgn(p + i);
if (i % 2) t = -t;
if (t != s)
{
n += 1;
s = t;
}
}
if (n % 2 == 1) return 1;
/* try to isolate one root */
return _fmpz_poly_num_real_roots(p, len) != 0;
}
int fmpz_poly_has_real_root(const fmpz_poly_t pol)
{
return _fmpz_poly_has_real_root(pol->coeffs, pol->length);
}