# Permutation polynomials of degree 8 over finite fields of odd characteristic

## TL;DR

This paper develops a computer algebra based algorithm to classify permutation polynomials of degree 8 over finite fields of odd characteristic, extending previous manual classifications to larger degrees.

## Contribution

It introduces an algorithmic approach using radicals of polynomial ideals to classify degree 8 permutation polynomials over finite fields, enabling results beyond manual analysis.

## Key findings

- Permutation polynomials of degree 8 exist over finite fields of order q if and only if q ≤ 31 and q ≠ 1 mod 8.
- The paper explicitly lists all such polynomials up to linear transformations.
- The method efficiently determines all degree 8 permutation polynomials over finite fields of odd order q > 8.

## Abstract

This paper provides an algorithmic generalization of Dickson's method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson's idea is to formulate from Hermite's criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most $6$ were essentially deduced from manual analysis of these polynomial equations. However, these polynomials, needed for that purpose when $d>6$, are too complicated to solve. Our idea is to make them more solvable by calculating some radicals of ideals generated by them, implemented by a computer algebra system (CAS). Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree $8$ over an arbitrary finite field of odd order $q>8$. The main result is that for an odd prime power $q>8$, a PP $f$ of degree $8$ exists over the finite field of order $q$ if and only if $q\leqslant 31$ and $q\not\equiv 1\ (\mathrm{mod}\ 8)$, and $f$ is explicitly listed up to linear transformations.