7 research outputs found
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
Case-finding of chronic obstructive pulmonary disease with questionnaire, peak flow measurements and spirometry : a cross-sectional study
Peer reviewedPublisher PD
Key Polynomials in dimension 2
Let be a two-dimensional regular local ring. In this paper, we prove that there is a bijection between the set of all valuations of centered at and valuations of centered at , where is the residue field of and and are independent variables. Moreover, we give a new proof, for the fact that the set of all normalized real valuations centered at admits a structure of non metric tree
On common extensions of valued fields
International audienceGiven a valuation on a field , an extension to an algebraic closure and an extension to . We want to study the common extensions of and to . First we give a detailed link between the minimal pairs notion and the key polynomials notion. Then we prove that in the case when 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 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 and study the relationship between them and key polynomials of Mac Lane -- Vaqui\'e. Associated to each abstract key polynomial , we define the truncation of with respect to and we study the properties of those truncations. Roughly speaking, is an approximation to defined by the key polynomial . We also define the notion of an abstract key polynomial being an \textbf{immediate successor} of another abstract key polynomial (in this situation we write ). The main comparison results proved in this paper are as follows:(1): An abstract key polynomial for is a Mac Lane -- Vaqui\'e key polynomial for the truncated valuation .(2): If are two abstract key polynomials for then is a Mac Lane -- Vaqui\'e key polynomial for . (3) which, for a monic polynomial and a valuation of , gives a sufficient condition for to be an abstract key polynomial for . Combined with an earlier result of M. Vaqui\'e, this describes a class of pairs of valuations such that is a Mac Lane -- Vaqui\'e key polynomial for and an abstract key polynomial for
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