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