Number of Equivalence Classes of Rational Functions over Finite Fields

Two rational functions $f,g\in {\mathbb{F}}_q(X)$ are said to be {\em equivalent} if there exist $\phi ,\psi \in {\mathbb{F}}_q(X)$ of degree one such that $g=\phi \circ f\circ \psi$. We give an explicit formula for the number of equivalence classes of rational functions of a given degree in ${\mathbb{F}}_q(X)$. This result should provide guidance for the current and future work on classifications of low degree rational functions over finite fields. We also determine the number of equivalence classes of polynomials of a given degree in ${\mathbb{F}}_q[X]$.