Very good and very bad field generators

Abstract

Let $\mathbf{k}$ be a field. A field generator is a polynomial $F\in \mathbf{k}[X,Y]$ satisfying $\mathbf{k}(F,G)=\mathbf{k}(X,Y)$ for some $G\in \mathbf{k}(X,Y)$. If $G$ can be chosen in $\mathbf{k}[X,Y]$, we call $F$ a good field generator; otherwise, $F$ is a bad field generator. These notions were first studied by Abhyankar, Jan, and Russell in the 1970s. The present paper introduces and studies the notions of “very good” and “very bad” field generators. We give theoretical results as well as new examples of bad and very bad field generators.