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CONTRACTING ENDOMORPHISMS OF VALUED FIELDS

Part of: Connections with logic Differential and difference algebra Model theory

Published online by Cambridge University Press:  08 May 2025

Yuval Dor ,
Yatir Halevi Open the ORCID record for Yatir Halevi [Opens in a new window]  and
With an appendix by itay Kaplan
Yuval Dor
Affiliation:
Apple Inc., Israel ([email protected])
Yatir Halevi*
Affiliation:
Department of Mathematics, University of Haifa, 199 Abba Khoushy Avenue, Haifa, Israel
*
[email protected]
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Abstract

We prove that the class of separably algebraically closed valued fields equipped with a distinguished Frobenius endomorphism $x \mapsto x^q$ is decidable, uniformly in q. The result is a simultaneous generalization of the work of Chatzidakis and Hrushovski (in the case of the trivial valuation) and the work of the first author and Hrushovski (in the case where the fields are algebraically closed).

The logical setting for the proof is a model completeness result for valued fields equipped with an endomorphism $\sigma $ which is locally infinitely contracting and fails to be onto. Namely, we prove the existence of a model complete theory $\widetilde {\mathrm {VFE}}$ amalgamating the theories $\mathrm {SCFE}$ and $\widetilde {\mathrm {VFA}}$ introduced in [5] and [11], respectively. In characteristic zero, we also prove that $\widetilde {\mathrm {VFE}}$ is NTP$_2$ and classify the stationary types: they are precisely those orthogonal to the fixed field and the value group.

Keywords

transformalvalued fieldsfrobeniusultraproductmodel complete

MSC classification

Primary: 12H10: Difference algebra 12L12: Model theory
Secondary: 03C60: Model-theoretic algebra
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 60
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

The second author was partially supported by ISF grants No. 555/21 and 290/19 and by the Fields Institute for Research in Mathematical Sciences. The author of the appendix would like to thank the Israel Science Foundation for their support of this research (grant no. 1254/18 and 804/22)

In memory of Zoé Chatzidakis, 1955–2025.

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