Permutations polynomials of the form G(X)k − L(X) and curves over finite fields

Abstract

For a positive integer k and a linearized polynomial L(X), polynomials of the form \(P(X)=G(X)^{k}-L(X) \in {\mathbb F}_{q^{n}}[X]\) are investigated. It is shown that when L has a non-trivial kernel and G is a permutation of \(\mathbb {F}_{q^{n}}\), then P(X) cannot be a permutation if \(\gcd (k,q^{n}-1)>1\). Further, necessary conditions for P(X) to be a permutation of \(\mathbb {F}_{q^{n}}\) are given for the case that G(X) is an arbitrary linearized polynomial. The method uses plane curves, which are obtained via the multiplicative and the additive structure of \(\mathbb {F}_{q^{n}}\), and their number of rational affine points.