Valuations on K[x] approaching a fixed irreducible polynomial

Abstract

For a fixed irreducible polynomial F we study the set VF of all valuations on K[x] bounded by valuations whose support is (F). The first main result presents a characterization for valuations in VF in terms of their graded rings. We also present a result which gives, for a fixed ν∈VF and a key polynomial Q∈KP(ν), the maximum value that augmented valuations in VF can assume on Q. This value is presented explicitly in terms of the slopes of the Newton polygon of F with respect to Q. Finally, we present some results about Artin-Schreier extensions that illustrate the applications that we have in mind for the results in this paper.