A group action on multivariate polynomials over finite fields

Let Fq be the finite field with q elements, where q is a power of a prime p. Recently, a particular action of the group GL2(Fq) on irreducible polynomials in Fq[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 Fq[x1,…,xn] and study the algebraic properties of the invariant elements.