On components of vectorial permutations of Fqn

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