A new criterion for the absolute irreducibility of multivariate polynomials over finite fields
Carlos Agrinsoni, Heeralal Janwa, Moises Delgado
TL;DR
The paper addresses the problem of testing absolute irreducibility of hypersurfaces defined by multivariate polynomials over finite fields, where existing criteria can be impractical. It introduces a gcd-based irreducibility criterion that only requires the leading form $F_d(\mathbf{X})$ to be square-free and the gcd $(F_d, F_{d-\gamma_1}, \dots, F_{d-\gamma_m})=1$, together with a degree-gap condition that $\gamma_m \notin \operatorname{span}_{\mathbb{N}}\{\gamma_1,\dots,\gamma_{m-1}\}$. The main theorem (Theorem 'heptanomials') shows $F(\mathbf{X})$ is absolutely irreducible under these hypotheses, and the paper extends this with a generalized degree-gap framework yielding bounds on the number of factors and criteria for absolute irreducibility in broader cases. The results generalize and subsume several known cases (binomials, trinomials, and certain quadrinomials) and have practical implications for APN conjectures, algebraic-geometric codes, and related cryptographic constructions. Because almost all polynomials are square-free, the method applies to a large class of polynomials, providing fast, gcd-based tests that avoid extension-field irreducibility checks.
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.