On components of vectorial permutations of [formula omitted]

Abstract

We consider vectorial mapsF(x1,…,xn)=(f1(x1,…,xn),…,fn(x1,…,xn)):Fqn↦Fqn, which induce permutations of Fqn. We show that the degrees of the components f1,f2,…,fn∈Fq[x1,…,xn] are at least 2 when 2≤deg(F)=d<q and d|(q−1). Our proof uses an absolutely irreducible curve over Fq and the number of rational points on it that we relate to the cardinality of the value set of a polynomial.