Key polynomials

Abstract

Let K→L be a field extension and ν a valuation of K. In order to study the possible extensions μ of ν to L (in the case of discrete ν of rank 1), S. MacLane introduced the notion of key polynomials, a certain well ordered subset of K[X], as well as the notion of augmented valuations, a way of building up μ by successive approximations. In a series of papers, M. Vaquié generalized MacLane’s notion of key polynomials to the case of arbitrary valuations ν (that is, valuations which are not necessarily discrete of rank 1). In the paper [3], Herrera Govantes, Olalla Acosta and Spivakovsky develop their own notion of key polynomials for extensions (K,ν)→(L,μ) of valued fields, where ν is of archimedean 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 Vaquié and F.J. Herrera Govantes, M.A. Olalla Acosta and M. Spivakovsky. Our main result is the comparison theorem (4.2). We also give an explicit example of a limit key polynomial in the case when ν and μ are centered in local noetherian rings.