Polynomial endomorphisms over finite fields: experimental results

TL;DR

This paper presents experimental results on polynomial endomorphisms over small finite fields, exploring properties like bijectivity and automorphisms for low degrees in three variables, and discusses interesting objects and conjectures found.

Contribution

It provides the first systematic computational study of polynomial endomorphisms over finite fields in three variables, highlighting properties and conjectures.

Findings

Identification of interesting polynomial endomorphisms

Discovery of potential new automorphisms and mock automorphisms

Formulation of conjectures based on computational data

Abstract

Given a finite field $\F_q$ and $n\in \N^*$, one could try to compute all polynomial endomorphisms $\F_q^n\lp \F_q^n$ up to a certain degree with a specific property. We consider the case $n=3$. If the degree is low (like 2,3, or 4) and the finite field is small ($q\leq 7$) then some of the computations are still feasible. In this article we study the following properties of endomorphisms: being a bijection of $\F_q^n\lp \F_q^n$, being a polynomial automorphism, being a {\em Mock automorphism}, and being a locally finite polynomial automorphism. In the resulting tables, we point out a few interesting objects, and pose some interesting conjectures which surfaced through our computations.