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Tropical functions on a skeleton
[Fonctions tropicales sur un squelette]
Antoine Ducros1 ; Ehud Hrushovski2 ; François Loeser3 ; Jinhe Ye4
1 Sorbonne Université, Université Paris-Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Campus Pierre et Marie Curie, case 247, 4 place Jussieu, 75252 Paris cedex 5, France
2 Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter Woodstock Road, Oxford OX2 6GG, UK
3 Institut universitaire de France, Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche CNRS, Campus Pierre et Marie Curie, case 247, 4 place Jussieu, 75252 Paris cedex 5, France
4 Mathematical Institute, University of Oxford, Andrew Wiles Building Radcliffe Observatory Quarter Woodstock Road, Oxford OX2 6GG, UK
Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 613-654.

Nous démontrons un résultat général de finitude pour le groupe abélien ordonné des fonctions tropicales sur un squelette dans l’analytifié de Berkovich d’une variété algébrique. Notre approche consiste à travailler dans le cadre des complétés stables de variétés algébriques, une version modèle théorique de l’analytification de Berkovich, pour lesquels nous démontrons un énoncé similaire dont notre résultat est une conséquence.

We prove a general finiteness statement for the ordered abelian group of tropical functions on skeleta in Berkovich analytifications of algebraic varieties. Our approach consists in working in the framework of stable completions of algebraic varieties, a model-theoretic version of Berkovich analytifications, for which we prove a similar result, of which the former one is a consequence.

Reçu le :
Accepté le :
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DOI : 10.5802/jep.261
Classification : 03C98, 14G22, 14T20
Keywords: Berkovich spaces, tropical geometry, skeleta, stable completion, Abhyankar valuations
Mot clés : Espaces de Berkovich, géométrie tropicale, squelettes, complété stable, valuations d’Abhyankar
Antoine Ducros 1 ; Ehud Hrushovski 2 ; François Loeser 3 ; Jinhe Ye 4

1 Sorbonne Université, Université Paris-Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Campus Pierre et Marie Curie, case 247, 4 place Jussieu, 75252 Paris cedex 5, France
2 Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter Woodstock Road, Oxford OX2 6GG, UK
3 Institut universitaire de France, Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche CNRS, Campus Pierre et Marie Curie, case 247, 4 place Jussieu, 75252 Paris cedex 5, France
4 Mathematical Institute, University of Oxford, Andrew Wiles Building Radcliffe Observatory Quarter Woodstock Road, Oxford OX2 6GG, UK
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Tropical functions on a skeleton},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Antoine Ducros; Ehud Hrushovski; François Loeser; Jinhe Ye. Tropical functions on a skeleton. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 613-654. doi : 10.5802/jep.261. https://jep.centre-mersenne.org/articles/10.5802/jep.261/

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