Augmented Valuation and Minimal Pair

2020/05/07 by Vaquié, Michel
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2005.03298

Abstract

Let (K, ν) be a valued field, the notions of augmented valuation, of limit augmented valuation and of admissible family of valuations enable to give a description of any valuation μ of K [x] extending ν. In the case where the field K is algebraically closed, this description is particularly simple and we can reduce it to the notions of minimal pair and pseudo-convergent family. Let (K, ν) be a henselian valued field and ν the unique extension of ν to the algebraic closure K of K and let μ be a valuation of K [x] extending ν, we study the extensions μ from μ to K [x] and we give a description of the valuations μ_i of K [x] which are the extensions of the valuations μ_i belonging to the admissible family associated with μ.

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