A group action on multivariate polynomials over finite fields

TL;DR

This paper extends a group action on univariate irreducible polynomials over finite fields to multivariate polynomials, exploring the algebraic properties of the resulting invariant elements.

Contribution

It introduces a natural extension of the group action of (_q) on multivariate polynomial rings over finite fields and studies their invariants.

Findings

Characterization of invariant polynomials under the extended group action

Analysis of algebraic properties of invariant elements

Foundation for further study of symmetries in multivariate polynomial rings

Abstract

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a power of a prime $p$. Recently, a particular action of the group $\mathrm{GL}_2(\mathbb F_q)$ on irreducible polynomials in $\mathbb F_q[x]$ has been introduced and many questions concerning the invariant polynomials have been discussed. In this paper, we give a natural extension of this action on the polynomial ring $\mathbb F_q[x_1, \ldots, x_n]$ and study the algebraic properties of the invariant elements.