Key Polynomials

The notion of key polynomials was first introduced in 1936 by S. Maclane in the case of discrete rank 1 valuations. . Let K -> L be a field extension and {\nu} a valuation of K. The original motivation for introducing key polynomials was the problem of describing all the extensions {\mu} of {\nu} to L. Take a valuation {\mu} of L extending the valuation {\nu}. In the case when {\nu} is discrete of rank 1 and L is a simple algebraic extension of K Maclane introduced the notions of key polynomials for {\mu} and augmented valuations and proved that {\mu} is obtained as a limit of a family of augmented valuations on the polynomial ring K[x]. In a series of papers, M. Vaqui\'e generalized MacLane's notion of key polynomials to the case of arbitrary valuations {\nu} (that is, valuations which are not necessarily discrete of rank 1). In the paper Valuations in algebraic field extensions, published in the Journal of Algebra in 2007, F.J. Herrera Govantes, M.A. Olalla Acosta and M. Spivakovsky develop their own notion of key polynomials for extensions (K, {\nu}) -> (L, {\mu}) of valued fields, where {\nu} is of archimedian rank 1 (not necessarily discrete) and give an explicit description of the limit key polynomials. Our purpose in this paper is to clarify the relationship between the two notions of key polynomials already developed by vaqui\'e and by F.J. Herrera Govantes, M.A. Olalla Acosta and M. Spivakovsky.Comment: arXiv admin note: text overlap with arXiv:math/0605193 by different author

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Key Polynomials in dimension 2

Let $R$ be a two-dimensional regular local ring. In this paper, we prove that there is a bijection between the set of all valuations of $Quot(R)$ centered at $R$ and valuations of $k(x,y)$ centered at $k[x,y]_{(x,y)}$, where $k$ is the residue field of $R$ and $x$ and $y$ are independent variables. Moreover, we give a new proof, for the fact that the set of all normalized real valuations centered at $R$ admits a structure of non metric tree

On common extensions of valued fields

International audienceGiven 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 minimal pairs notion and the key polynomials notion. Then we prove that in the case when $w$ is a transcendental extension, then any sequence of key polynomials admits a maximal element, and in case this sequence does not contain a limit key polynomial, then any root of the last key polynomial, describe a common extension

Abstract key polynomials and comparison theorems with the key polynomials of Mac Lane -- Vaquie

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 polynomials. In this paper we define a related notion of \textbf{abstract key polynomials} associated to $\iota$ and study the relationship between them and key polynomials of Mac Lane -- Vaqui\'e. Associated to each abstract key polynomial $Q$, we define the truncation $\mu_{Q}$ of $\mu$ with respect to $Q$ and we study the properties of those truncations. Roughly speaking, $\mu_{Q}$ is an approximation to $\mu$ defined by the key polynomial $Q$. We also define the notion of an abstract key polynomial $Q'$ being an \textbf{immediate successor} of another abstract key polynomial $Q$ (in this situation we write $Q<Q'$). The main comparison results proved in this paper are as follows:(1): An abstract key polynomial for $\mu$ is a Mac Lane -- Vaqui\'e key polynomial for the truncated valuation $\mu_{Q}$.(2): If $Q<Q'$ are two abstract key polynomials for $\mu$ then $Q'$ is a Mac Lane -- Vaqui\'e key polynomial for $\mu_{Q}$. (3) which, for a monic polynomial $Q\in K[x]$ and a valuation $\mu'$ of $K(x)$, gives a sufficient condition for $Q$ to be an abstract key polynomial for $\mu'$. Combined with an earlier result of M. Vaqui\'e, this describes a class of pairs of valuations $(\mu,\mu')$ such that $Q$ is a Mac Lane -- Vaqui\'e key polynomial for $\mu$ and an abstract key polynomial for $\mu'$

Key polynomials for simple extensions of valued fields

In this paper we present a refined version of MacLane's theory of key polynomials [16]-[17], similar to those considered by M. Vaquié [24]-[27], and reminiscent of related objects studied by Abhyankar and Moh (approximate roots [1], [2]) and T.C. Kuo [14], [15].Let (K, ν_0) be a valued field. Given a simple transcendental extension of valued fields ι : K → K(x) we associate to ι a countable well ordered set of polynomials of K[x] called key polynomials. We define limit key polynomials and give an explicit description of them. We show that the order type of the set of key polynomials is bounded by ω × ω. If char k_ν_0 = 0 and rk ν_0 = 1, the order type is bounded by ω + 1