Incidence bounds in positive characteristic via valuations and distality
[Bornes d’incidence en caractéristique positive par les valuations et la distalité]
Bays, Martin1 ; Martin, Jean-François2
1 Mathematisches Institut und Institut für Mathematische Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstrasse 62, 48149 Muenster, Germany
2 Université Paris-Saclay, Paris, France
Annales Henri Lebesgue, Tome 6 (2023), pp. 627-641

We prove distality of quantifier-free relations on valued fields with finite residue field. By a result of Chernikov–Galvin–Starchenko, this yields Szemerédi–Trotter-like incidence bounds for function fields over finite fields. We deduce a version of the Elekes–Szabó theorem for such fields.

Nous démontrons la distalité des relations sans quantificateur sur les corps valués de corps résiduel fini. D’après un résultat de Chernikov–Galvin–Starchenko, cela implique des bornes d’incidence de type Szemerédi–Trotter pour les corps de fonctions sur les corps finis. Nous en déduisons une version du théorème d’Elekes–Szabó pour de tels corps.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/ahl.174
Classification : 03C98, 03C45, 05D99
Keywords: Szemerédi-Trotter, incidence bounds, distality, Elekes-Szabó

Bays, Martin 1 ; Martin, Jean-François 2

1 Mathematisches Institut und Institut für Mathematische Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstrasse 62, 48149 Muenster, Germany
2 Université Paris-Saclay, Paris, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Bays, Martin; Martin, Jean-François. Incidence bounds in positive characteristic via valuations and distality. Annales Henri Lebesgue, Tome 6 (2023), pp. 627-641. doi: 10.5802/ahl.174

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