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

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.

Keywords:

• Irreducible polynomials
• prime numbers
• zero-free sectors

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 11R09
• Secondary: 11C08