In this paper we establish the relation between key polynomials (as defined in [12]) and
minimal pairs of definition of a valuation. We also discuss truncations of valuations on a …

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 …

One of the main goals of this paper is to present the relation between limit key polynomials
and MacLane-Vaquié limit key polynomials. This is a continuation of the work started in [3] …

Let ν be a valuation of arbitrary rank on the polynomial ring K [x] with coefficients in a field K.
We prove comparison theorems between MacLane–Vaquié key polynomials for valuations …

Let $\iota: K\hookrightarrow L\cong K (x) $ be a simple transcendental extension of valued
fields, where $ K $ is equipped with a valuation $\nu $ of rank 1. That is, we assume given a …

We build on the correspondence between abstract key polynomials and minimal pairs made
by Novacoski and show how to relate the valuations that are generated by each object. We …

In this paper, we study the structure of the graded ring associated to a limit key polynomial Q
n in terms of the key polynomials that define Q n. In order to do that, we use direct limits. In …

In this paper, we extend the theory of minimal limit key polynomials of valuations on the
polynomial ring $\kx $. We use the theory of cuts on ordered abelian groups to show that the …

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 …

Suppose that (K, $\nu $) is a valued field, f (z) $\in $ K [z] is a unitary and irreducible
polynomial and (L, $\omega $) is an extension of valued fields, where L= K [z]/(f (z)). Further …