Cubic rational expressions over a finite field

TL;DR

This paper classifies cubic rational expressions over finite fields, providing a complete classification for even fields and an upper bound for odd fields, advancing understanding of their structural properties.

Contribution

It offers a full classification of cubic rational expressions over finite fields when the field size is even, and establishes an upper bound for the number of classes when the field size is odd.

Findings

Complete classification for even finite fields

Upper bound of 4q for odd finite fields

Advances understanding of rational expressions over finite fields

Abstract

We study and partially classify cubic rational expressions $g(x)/h(x)$ over a finite field $\mathbb{F}_q$, up to pre- and post-composition with independent M\"obius transformations. In particular, we obtain a full classification when $q$ is even, and prove an upper bound of $4q$ for the number of equivalence classes when $q$ is odd.