Tropical functions on a skeleton
[Fonctions tropicales sur un squelette]
Ducros, Antoine1 ; Hrushovski, Ehud2 ; Loeser, François3 ; Ye, Jinhe4
1Sorbonne 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
2Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter Woodstock Road, Oxford OX2 6GG, UK
3Institut 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
4Mathematical 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

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.261
Classification : 03C98, 14G22, 14T20
Keywords: Berkovich spaces, tropical geometry, skeleta, stable completion, Abhyankar valuations
Mots-clés : Espaces de Berkovich, géométrie tropicale, squelettes, complété stable, valuations d’Abhyankar

Ducros, Antoine  1   ; Hrushovski, Ehud  2   ; Loeser, François  3   ; Ye, Jinhe  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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Ducros, Antoine; Hrushovski, Ehud; Loeser, François; Ye, Jinhe. Tropical functions on a skeleton. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 613-654. doi: 10.5802/jep.261

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