Very good and very bad field generators

TL;DR

This paper explores the concepts of very good and very bad field generators in polynomial rings over fields, providing new theoretical insights and examples to deepen understanding of their properties.

Contribution

It introduces the notions of very good and very bad field generators and offers new theoretical results and examples distinguishing these classes.

Findings

Introduction of very good and very bad field generators

New examples of bad and very bad field generators

Theoretical results characterizing these generators

Abstract

Let k be a field. A "field generator" is a polynomial F in k[X,Y] satisfying k(F,G) = k(X,Y) for some G in k(X,Y). If G can be chosen in 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.