Abstract key polynomials and comparison theorems with the key polynomials of Mac Lane–Vaquié

Abstract

Let $(K,\nu)$ be a valued field and $K(x)$ a simple purely transcendental extension of $K$. In the nineteen thirties, in order to study the possible extensions of $\nu $ to $K(x)$, S. Mac Lane considered the special case when $\nu $ is discrete of rank $1$, and introduced the notion of key polynomials. M. Vaquié extended this definition to the case of arbitrary valuations. In this paper we give a new definition of key polynomials (which we call abstract key polynomials) and study the relationship between them and key polynomials of Mac Lane–Vaquié.