Permutation polynomials, fractional polynomials, and algebraic curves

TL;DR

This paper proves a conjecture on permutation trinomials over finite fields, introduces new permutation polynomial families, and analyzes permutation quadrinomials using algebraic curves related to fractional polynomials.

Contribution

It confirms a conjecture on permutation trinomials and extends the classification of permutation polynomials over finite fields with new examples and generalizations.

Findings

Proved a conjecture on permutation trinomials over ^{2k}.

Presented new families of permutation polynomials over ^k and ^k.

Analyzed permutation quadrinomials using algebraic curves associated with fractional polynomials.

Abstract

In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over $\mathbb{F}_3^{2k}$. In addition, new examples and generalizations of some families of permutation polynomials of $\mathbb{F}_{3^k}$ and $\mathbb{F}_{5^k}$ are given. We also study permutation quadrinomials of type $Ax^{q(q-1)+1} + Bx^{2(q-1)+1} + Cx^{q} + x$. Our method is based on the investigation of an algebraic curve associated with a {fractional polynomial} over a finite field.