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

For a positive integer k and a linearized polynomial L(X), polynomials of the form P(X) = G(X)(k) - L(X) is an element of F-qn [X] are investigated. It is shown that when L has a non-trivial kernel and G is a permutation of F-qn, 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 F-qn 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 F-qn, and their number of rational affine points.