Complete permutation polynomials from exceptional polynomials

TL;DR

This paper classifies specific complete permutation polynomials over finite fields and explores indecomposable exceptional polynomials of degrees 8 and 9, advancing understanding of polynomial structures in finite field theory.

Contribution

It provides a classification of complete permutation polynomials of a particular form for certain prime conditions and degree cases, and studies indecomposable exceptional polynomials of degrees 8 and 9.

Findings

Classified complete permutation polynomials for prime-related conditions.

Identified indecomposable exceptional polynomials of degrees 8 and 9.

Extended understanding of polynomial structures over finite fields.

Abstract

We classify complete permutation polynomials of type $aX^{\frac{q^n-1}{q-1}+1}$ over the finite field with $q^n$ elements, for $n+1$ a prime and $n^4 < q$. For the case $n+1$ a power of the characteristic we study some known families. We also classify indecomposable exceptional polynomials of degree $8$ and $9$.