Vol. 319, No. 1, 2022

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 328: 1
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
Rigidity of valuative trees under henselization

Enric Nart

Vol. 319 (2022), No. 1, 189–211

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.

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