FLINT3.7.0-dev
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_fmpz_poly_bound_roots, fmpz_poly_bound_roots

Synopsis#include <flint/fmpz_poly.h>
void _fmpz_poly_bound_roots(fmpz_t bound, const fmpz * poly, slong len)source
void fmpz_poly_bound_roots(fmpz_t bound, const fmpz_poly_t poly)source

Set bound to a nonnegative integer that is intended to bound the absolute values of all complex roots, computed from Fujiwara's bound [Fuj1916]

2max⁡(∣an−1an∣,∣an−2an∣1/2,…,∣a1an∣1/(n−1),∣a02an∣1/n)2 \max \left( \left|\frac{a_{n-1}}{a_n}\right|, \left|\frac{a_{n-2}}{a_n}\right|^{1/2}, \ldots, \left|\frac{a_1}{a_n}\right|^{1/(n-1)}, \left|\frac{a_0}{2 a_n}\right|^{1/n} \right)

for f=∑i=0naixif = \sum_{i=0}^n a_i x^i, with each term rounded up to an integer (for n≥2n \ge 2, as ⌊⋅⌋+1\lfloor \cdot \rfloor + 1). The bound is zero for constant polynomials.

References#

  • [Fuj1916] M. Fujiwara. Über die obere Schranke des absoluten Betrages der Wurzeln einer algebraischen Gleichung. Tôhoku Mathematical Journal 10 (1916), 167–171.
Cite this page

The FLINT team, “_fmpz_poly_bound_roots, fmpz_poly_bound_roots (fmpz_poly.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/fmpz_poly/_fmpz_poly_bound_roots.html (generated 2026-11-02).