
TL;DR
This paper compares two different notions of key polynomials in valuation theory, clarifying their relationship and extending understanding of valuations in algebraic field extensions.
Contribution
It clarifies the relationship between Vaquié's and Herrera Govantes et al.'s notions of key polynomials in valuation theory.
Findings
Established the connection between two key polynomial frameworks.
Extended the description of limit key polynomials.
Enhanced understanding of valuation extensions in algebraic fields.
Abstract
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,…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory
