A new construction of permutation polynomials over $\mathbb{F}_{q^3}$

Abstract

We determine all permutation polynomials among several families of polynomials over $\mathbb{F}_{q^3}$ for arbitrary prime powers $q$. We obtain some new families of permutation polynomials over $\mathbb{F}_{q^3}$ with simple coefficients for infinitely many characteristics. As a specific consequence, our results resolve the generalization of conjectures of Zhang, Zheng, Wang, Peng, and Li in the even characteristic. Our proofs are conceptually short and involve no complicated computations, in contrast to the proofs of results on permutation polynomials which were published previously. Moreover, we develop a totally new systematic method in this paper for the study of permutation polynomials.