Key polynomials for simple extensions of valued fields

Abstract

In this paper we present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaqui\'e and reminiscent of approximate roots of Abhyankar and Moh. Given a simple transcendental extension of valued fields, we associate to it a countable well-ordered set of polynomials called key polynomials. We define limit key polynomials and give explicit formulae for them. We give an explicit bound on the order type of the set of key polynomials.