A classification of permutation polynomials of degree $7$ over finite fields

TL;DR

This paper provides a comprehensive classification of permutation polynomials of degree 7 over finite fields, utilizing computational tools to systematically analyze their structure across all odd prime power fields.

Contribution

It offers the first complete classification of degree 7 permutation polynomials over finite fields, extending previous partial results and employing SageMath for verification.

Findings

Complete classification of degree 7 permutation polynomials over finite fields.

Identification of all such polynomials up to linear transformations.

Use of computational algebra system SageMath for analysis.

Abstract

Up to linear transformations, we give a classification of all permutation polynomials of degree $7$ over $\mathbb{F}_{q}$ for any odd prime power $q$, with the help of the SageMath software.