Field generators in two variables and birational endomorphisms of $\mathbb{A}^2$

Abstract

This article is a survey of two subjects: the first part is devoted to field generators in two variables, and the second to birational endomorphisms of the affine plane. Each one of these subjects originated in Abhyankar's seminar in Purdue University in the 1970s. Note that the part on field generators is more than a survey, since it contains a considerable amount of new material.