A note on permutation polynomials over finite fields

TL;DR

This paper resolves two recent conjectures on permutation polynomials over finite fields and introduces a new class of permutation trinomials that generalize previous examples, advancing the understanding of their structure and applications.

Contribution

It settles two open conjectures and presents a novel class of permutation trinomials involving trace functions, expanding the known families of permutation polynomials.

Findings

Two conjectures on permutation polynomials are proved.

A new class of permutation trinomials is introduced.

The new class generalizes existing examples of permutation polynomials.

Abstract

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are settled. Moreover, a new class of permutation trinomials of the form $x+\gamma \textup{Tr}_{q^n/q}(x^k)$ is also presented, which generalizes two examples of [10].