A new criterion for the absolute irreducibility of multivariate polynomials over finite fields

TL;DR

This paper introduces a new criterion for determining the absolute irreducibility of multivariate polynomials over finite fields, simplifying the process by avoiding extensive irreducibility tests and relying on GCD computations and the square-free property.

Contribution

The authors propose a novel criterion that determines absolute irreducibility without testing in extension fields, based solely on GCD and square-free conditions.

Findings

Criterion is valid for almost all multivariate polynomials

Does not require irreducibility testing in extension fields

Relies on GCD computations and square-free property

Abstract

A key property of an algebraic variety is whether it is absolutely irreducible, meaning that it remains irreducible over the algebraic closure of its defining field, and determining absolute irreducibility is important in algebraic geometry and its applications in coding theory, cryptography, and other fields. Among the applications of absolute irreducibility are bounding the number of rational points via the Weil conjectures and establishing exceptional APN and permutation properties of functions over finite fields. In this article, we present a new criterion for the absolute irreducibility of hypersurfaces defined by multivariate polynomials over finite fields. Our criterion does not require testing for irreducibility in the ground or extension fields, assuming that the leading form is square-free. We just require multivariate GCD computations and the square-free property. Since almost all polynomials are known to be square-free, our absolute irreducibility criterion is valid for almost all multivariate polynomials.