Vol. 319, No. 1, 2022

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Rigidity of valuative trees under henselization

Enric Nart

Vol. 319 (2022), No. 1, 189–211
Abstract

Let (K,v) be a valued field and let (Kh,vh) be the henselization determined by the choice of an extension of v to an algebraic closure of K. Consider an embedding v(K)Λ of the value group into a divisible ordered abelian group. Let 𝒯 (K,Λ), 𝒯 (Kh,Λ) be the trees formed by all Λ-valued extensions of v, vh to the polynomial rings K[x], Kh[x], respectively. We show that the natural restriction mapping 𝒯 (Kh,Λ) 𝒯 (K,Λ) is an isomorphism of posets.

As a consequence, the restriction mapping 𝒯vh 𝒯v is an isomorphism of posets too, where 𝒯v, 𝒯vh are the trees whose nodes are the equivalence classes of valuations on K[x], Kh[x] whose restrictions to K, Kh are equivalent to v, vh, respectively.

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Keywords
henselization, key polynomial, valuation, valuative tree
Mathematical Subject Classification
Primary: 13A18
Secondary: 12J20, 13J10, 14E15
Milestones
Received: 4 February 2022
Revised: 15 April 2022
Accepted: 16 April 2022
Published: 28 August 2022
Authors
Enric Nart
Departament de Matemàtiques
Universitat Autònoma de Barcelona
Cerdanyola del Vallès
Spain