Minimal limit key polynomials

Abstract

In this paper, we extend the theory of minimal limit key polynomials of valuations on the polynomial ring $K\left[x\right]$. Minimal key polynomials are useful to describe, for instance, the defect of an extension of valued fields. We use the theory of cuts on ordered abelian groups to show that the previous results on bounded sets of key polynomials of rank one valuations extend to vertically bounded sets of key polynomials of valuations of an arbitrary rank. We also discuss properties of minimal limit key polynomials in the vertically unbounded case.