Permutation and local permutation polynomial of maximum degree

TL;DR

This paper explores permutation and local permutation polynomials over multivariate polynomial rings in finite fields, constructing maximum degree examples and extending previous theoretical results in the field.

Contribution

It introduces new constructions of maximum degree permutation and local permutation polynomials in multivariate polynomial rings over finite fields.

Findings

Constructed permutation polynomials of degree n(q-1)-1

Constructed local permutation polynomials of degree n(q-2) for q>3

Extended known results on permutation polynomials in multiple variables

Abstract

Let $F_q$ be the finite field with $q$ elements and $F_q[x_1,\ldots, x_n]$ the ring of polynomials in $n$ variables over $F_q$. In this paper we consider permutation polynomials and local permutation polynomials over $F_q[x_1,\ldots, x_n]$, which define interesting generalizations of permutations over finite fields. We are able to construct permutation polynomials in $F_q[x_1,\ldots, x_n]$ of maximum degree $n(q-1)-1$ and local permutation polynomials in $F_q[x_1,\ldots, x_n]$ of maximum degree $n(q-2)$ when $q>3$, extending previous results.