Français
Tropical functions on skeletons
Multi angle
Authors : Ducros, Antoine (Author of the conference)
CIRM (Publisher )
03C98 - Applications of model theory
14G22 - Rigid analytic geometry
14T20
CIRM (Publisher )
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Abstract :
Skeletons are subsets of non-archimedean spaces (in the sense of Berkovich) that inherit from the ambiant space a natural PL (piecewise-linear) structure, and if $S$ is such a skeleton, for every invertible holomorphic function $f$ defined in a neighborhood of $S$, the restriction of $\log |f|$ to $S$ is $\mathrm{PL}$.In this talk, I will present a joint work with E. Hrushovski, F. Loeser and J. Ye in which we consider an irreducible algebraic variety $X$ over an algebraically closed, non-trivially valued and complete non-archimedean field $k$, and a skeleton $S$ of the analytification of $X$ defined using only algebraic functions, and consisting of Zariski-generic points. If $f$ is a non-zero rational function on $X$ then $\log |f|$ indices a $\mathrm{PL}$ function on $S$, and if we denote by $E$ the group of all $\mathrm{PL}$ functions on $S$ that are of this form, we prove the following finiteness result on the group $E$ : it is stable under min and max, and there exist finitely many non-zero rational functions $f_1, \ldots, f_m$ on $X$ such that $E$ is generated, as a group equipped with min and max operators, by the $\log \left|f_i\right|$ and the constants $|a|$ for a in $k^*$. Our proof makes a crucial use of Hrushovski-Loesers theory of stable completions, which are model-theoretic avatars of Berkovich spaces.
Keywords : Berkovich spaces; skeletons; tropical functions
MSC Codes : Keywords : Berkovich spaces; skeletons; tropical functions
03C98 - Applications of model theory
14G22 - Rigid analytic geometry
14T20
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 23/06/2023 Conference Date : 29/05/2023 Subseries : Research talks arXiv category : Algebraic Geometry ; Logic Mathematical Area(s) : Algebraic & Complex Geometry ; Logic and Foundations Format : MP4 (.mp4) - HD Video Time : 00:57:48 Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2023-05-29_Ducros_1.mp4 
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Information on the Event
Event Title : Model theory of valued fields / Théorie des modèles des corps valuésEvent Organizers : Chatzidakis, Zoé ; Jahnke, Franziska ; Rideau-Kikuchi, Silvain Dates : 29/05/2023 - 02/06/2023 Event Year : 2023 Event URL : https://conferences.cirm-math.fr/2761.html
Citation Data
DOI : 10.24350/CIRM.V.20051403Cite this video as: Ducros, Antoine (2023). Tropical functions on skeletons. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20051403 URI : http://dx.doi.org/10.24350/CIRM.V.20051403 |
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Bibliography
- DUCROS, Antoine, HRUSHOVSKI, Ehud, LOESER, François, et al. Tropical functions on a skeleton. arXiv preprint arXiv:2210.04003, 2022. - https://doi.org/10.48550/arXiv.2210.04003
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