Cubic rational expressions over a finite field

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 Möbius 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.

24 pages; improved bound in Section 6 (lowered from 6q-2 to 4q), extended to include char 3; consequent rearrangement and renumbering of some theorems; discussion of related literature on cubic extensions included in Section 2