A group action on multivariate polynomials over finite fields

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 ${\mathbb{F}}_q[x_1,\dots ,x_n]$ and study the algebraic properties of the invariant elements.

29 pages