Permutation polynomials and complete permutation polynomials over $\mathbb{F}_{q^{3}}$

TL;DR

This paper introduces new classes of permutation and complete permutation polynomials over the finite field _{q^3}, expanding the understanding of polynomial permutations in higher-degree finite fields.

Contribution

It develops novel constructions of permutation polynomials over _{q^3} using multivariate methods and resultants, including some that are complete mappings.

Findings

Several new classes of sparse permutation polynomials over _{q^3}

Construction of some complete permutation polynomials

Extension of permutation polynomial theory to _{q^3}

Abstract

Motivated by many recent constructions of permutation polynomials over $\mathbb{F}_{q^2}$, we study permutation polynomials over $\mathbb{F}_{q^3}$ in terms of their coefficients. Based on the multivariate method and resultant elimination, we construct several new classes of sparse permutation polynomials over $\mathbb{F}_{q^3}$, $q=p^{k}$, $p\geq3$. Some of them are complete mappings.