Related papers: Key Polynomials

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 rank 1 valuation $\nu$ of $K$ and its extension $\nu'$ to…

In this paper we give an introduction on how one can extend a valuation from a field $K$ to the polynomial ring $K[x]$ in one variable over $K$. This follows a similar line as the one presented by the author in his talk at ALaNT 5. We will…

Let $K\to L$ be an algebraic field extension and $\nu$ a valuation of $K$. The purpose of this paper is to describe the totality of extensions $\left\{\nu'\right\}$ of $\nu$ to $L$ using a refined version of MacLane's key polynomials. In…

In this paper we introduce a new concept of key polynomials for a given valuation $\nu$ on $K[x]$. We prove that such polynomials have many of the expected properties of key polynomials as those defined by MacLane and Vaqui\'e, for…

In this paper we present characterizations of the sets of key polynomials and abstract key polynomials for a valuation $\mu$ of $K(x)$, in terms of (ultrametric) balls in the algebraic closure $\overline K$ of $K$ with respect to $v$, a…

Let $\iota:(K,\nu)\hookrightarrow(K(x),\mu)$ be a simple purely transcendental extension of valued fields. In order to study such an extension, M. Vaqui\'e, generalizing an earlier construction of S. Mac Lane, introduced the notion of Key…

We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove…

In this paper we establish the relation between key polynomials (as defined in \cite{SopivNova}) and minimal pairs of definition of a valuation. We also discuss truncations of valuations on a polynomial ring $K[x]$. We prove that a…

The main goal of this paper is to characterize limit key polynomials for a valuation $\nu$ on $K[x]$. We consider the set $\Psi_\alpha$ of key polynomials for $\nu$ of degree $\alpha$. We set $p$ be the exponent characteristic of $\nu$. Our…

Let $\nu$ be a valuation of arbitrary rank on the polynomial ring $K[x]$ with coefficients in a field $K$. We prove comparison theorems between MacLane-Vaqui\'e key polynomials for valuations $\mu\le\nu$ and abstract key polynomials for…

One of the main goals of this paper is to present the relation of limit key polynomials and limit MacLane-Vaqui\'e key polynomials. This is a continuation of the work started in a paper by Decaup, Mahboub and Spivakovsky, where it is proved…

Let K be a field. For a given valuation on K[x], we determine the structure of its graded algebra and describe its set of key polynomials, in terms of any given key polynomial of minimal degree. We also characterize valuations not admitting…

This article is a natural construction of our previous works. In this article, we employ similar ideas due to MacLane to provide an estimate of IC(K(X)|K,v) when (K(X)|K,v) is a valuation algebraic extension. Our central result is an…

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 previous results on bounded sets of key polynomials of…

Suppose that $(K,v_0)$ is a valued field, $f(x)\in K[x]$ is a monic and irreducible polynomial and $(L,v)$ is an extension of valued fields, where $L=K[x]/(f(x))$. Let $A$ be a local domain with quotient field $K$ dominated by the valuation…

Let $K$ be a field equipped with a discrete valuation $v$. In a pioneering work, S. MacLane determined all valuations on $K(x)$ extending $v$. His work was recently reviewed and generalized by M. Vaqui\'e, by using the graded algebra of a…

In this paper, for a valued field $(K, v)$ of arbitrary rank and an extension $w$ of $v$ to $K(X),$ a relation between induced complete sequences of abstract key polynomials and MacLane-Vaqui\'e chains is given.

Let $(K,v)$ be a valued field. We review some results of MacLane and Vaqui\'e on extensions of $v$ to valuations on the polynomial ring $K[x]$. We introduce certain MacLane-Vaqui\'e chains of residually transcendental valuations, and we…

Given a valuation $v$ on a field $K$, an extension $\bar{v}$ to an algebraic closure and an extension $w$ to $K(X)$. We want to study the common extensions of $\bar{v}$ and $w$ to $\bar{K}(X)$. First we give a detailed link between the…

An exponential polynomial is a finite linear sum of terms $P(z)e^{Q(z)}$, where $P(z)$ and $Q(z)$ are polynomials. The early results on the value distribution of exponential polynomials can be traced back to Georg P\'olya's paper published…