More on the DLW conjectures

Abstract

We prove two conjectures involving permutation polynomials in a paper of Dmytrenko, Lazebnik, Williford, in a low degree regime, using the theory of algebraic curves over finite fields. More precisely, we prove that Conjecture A holds whenever q≥max⁡{(2k−1)2+1,1.823(4k2−14k+12)}, whereas Conjecture B holds if q≥2.233(9k2−21k+12). Although one of these conjectures was already proved by Hou without any restriction on the degree of the polynomials, we consider the proof contained in this paper is more direct and less computational.