Zero-free angular sectors and lens-shaped regions for polynomials, with applications to irreducibility

TL;DR

This paper introduces explicit zero-free regions in the complex plane for real polynomials, which are used to derive new irreducibility criteria for polynomials with integer coefficients based on prime values.

Contribution

The work provides explicit zero-free angular sectors and lens-shaped regions for polynomials, linking geometric properties to irreducibility criteria.

Findings

Explicit zero-free sectors depend only on polynomial degree

Zero-free lens-shaped regions relate to reciprocal polynomials

Criteria for irreducibility based on prime or prime power values

Abstract

For a real polynomial $f$ we present explicit zero-free angular sectors in the complex plane, symmetric with respect to the real axis, with angles depending only on the degree of $f$, and vertices expressed in terms of the coefficients of $f$. We also describe zero-free lens-shaped regions for $f$ that are associated to the zero-free sectors of the reciprocal of $f$. As an application, we use these zero-free regions to obtain irreducibility criteria for polynomials with integer coefficients that take a prime or a prime power value.