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

TL;DR

This paper introduces a new systematic method for constructing and identifying permutation polynomials over finite fields, resolving several conjectures and providing infinitely many new examples with simple coefficients.

Contribution

It develops a novel, systematic approach for studying permutation polynomials over f_{q^3}, generalizes existing conjectures, and simplifies the proof process.

Findings

Identified all permutation polynomials in certain families over f_{q^3} for any prime power q.

Discovered new families of permutation polynomials with simple coefficients for infinitely many q.

Resolved conjectures of Zhang, Zheng, Wang, Peng, and Li in even characteristic.

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.