Vol. 311, No. 1, 2021

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MacLane–Vaquié chains of valuations on a polynomial ring

Enric Nart

Vol. 311 (2021), No. 1, 165–195
DOI: 10.2140/pjm.2021.311.165
Abstract

Let (K,v) be a valued field. We reinterpret some results of MacLane and Vaquié on extensions of v to valuations on the polynomial ring K[x]. We introduce certain MacLane–Vaquié chains constructed as a mixture of ordinary and limit augmentations of valuations. Every valuation ν on K[x] is a limit (in a certain sense) of a countable MacLane–Vaquié chain. This chain underlying ν is essentially unique and contains discrete arithmetic data yielding an explicit description of the graded algebra of ν as an algebra over the graded algebra of v.

Keywords
graded algebra, key polynomial, MacLane–Vaquié chain, valuation
Mathematical Subject Classification
Primary: 13A18
Secondary: 12J20
Milestones
Received: 14 September 2020
Revised: 18 December 2020
Accepted: 8 January 2021
Published: 17 March 2021
Authors
Enric Nart
Departament de Matemàtiques
Universitat Autònoma de Barcelona
Cerdanyola del Vallès
Catalonia
Spain