Volume 6 (2023)
Bays, Martin;  Martin, Jean-François
Incidence bounds in positive characteristic via valuations and distality
Annales Henri Lebesgue, Volume 6 (2023), pp. 627-641.

Metadata

DOI10.5802/ahl.174
Keywords Szemerédi-Trotter, incidence bounds, distality, Elekes-Szabó

Abstract

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.


References

[ACGZ22] Aschenbrenner, Matthias; Chernikov, Artem; Gehret, Allen; Ziegler, Martin Distality in valued fields and related structures, Trans. Am. Math. Soc., Volume 375 (2022) no. 7, pp. 4641-4710 | DOI | MR | Zbl

[BB21] Bays, Martin; Breuillard, Emmanuel Projective geometries arising from Elekes–Szabo problems, Ann. Sci. Éc. Norm. Supér., Volume 54 (2021) no. 3, pp. 627-681 | DOI | MR | Zbl

[BKT04] Bourgain, J.; Katz, N.; Tao, Terence A sum-product estimate in finite fields, and applications, Geom. Funct. Anal., Volume 14 (2004) no. 1, pp. 27-57 | DOI | MR | Zbl

[CGS20] Chernikov, Artem; Galvin, David; Starchenko, Sergei Cutting lemma and Zarankiewicz’s problem in distal structures, Sel. Math., New Ser., Volume 26 (2020) no. 2, 25 | DOI | MR | Zbl

[CPS] Chernikov, Artem; Peterzil, Ya’acov; Starchenko, Sergei Model-theoretic Elekes–Szabó for stable and o-minimal hypergraphs (https://arxiv.org/abs/2104.02235v1)

[CS15] Chernikov, Artem; Simon, Pierre Externally definable sets and dependent pairs II, Trans. Am. Math. Soc., Volume 367 (2015) no. 7, pp. 5217-5235 | DOI | MR | Zbl

[CS21] Chernikov, Artem; Starchenko, Sergei Model-theoretic Elekes–Szabó in the strongly minimal case, J. Math. Log., Volume 21 (2021) no. 2 | DOI | MR | Zbl

[EH91] Evans, David M.; Hrushovski, Ehud Projective planes in algebraically closed fields, Proc. Lond. Math. Soc., Volume 62 (1991) no. 1, pp. 1-24 | DOI | MR | Zbl

[EP05] Engler, Antonio J.; Prestel, Alexander Valued fields, Springer Monographs in Mathematics, Springer, 2005 | MR | Zbl

[ES12] Elekes, György; Szabó, Endre How to find groups? (and how to use them in Erdös geometry?), Combinatorica, Volume 32 (2012) no. 5, pp. 537-571 | DOI | MR | Zbl

[FPS + 17] Fox, Jacob; Pach, János; Sheffer, Adam; Suk, Andrew; Zahl, Joshua A semi-algebraic version of Zarankiewicz’s problem, J. Eur. Math. Soc., Volume 19 (2017) no. 6, pp. 1785-1810 | DOI | MR | Zbl

[Hol95] Holly, Jan E. Canonical forms for definable subsets of algebraically closed and real closed valued fields, J. Symb. Log., Volume 60 (1995) no. 3, pp. 843-860 | DOI | MR | Zbl

[Hru13] Hrushovski, Ehud On pseudo-finite dimensions, Notre Dame J. Formal Logic, Volume 54 (2013) no. 3-4, pp. 463-495 | DOI | MR | Zbl

[KSW11] Kaplan, Itay; Scanlon, Thomas; Wagner, Frank O Artin–Schreier extensions in NIP and simple fields, Isr. J. Math., Volume 185 (2011) no. 1, pp. 141-153 | DOI | MR | Zbl

[Lan02] Lang, Serge Algebra, Graduate Texts in Mathematics, 211, Springer, 2002 | DOI | MR | Zbl

[SdZ17] Stevens, Sophie; de Zeeuw, Frank An improved point-line incidence bound over arbitrary fields, Bull. Lond. Math. Soc., Volume 49 (2017) no. 5, pp. 842-858 | DOI | MR | Zbl

[Sim20] Simon, Pierre Type decomposition in NIP theories, J. Eur. Math. Soc., Volume 22 (2020) no. 2, pp. 455-476 | DOI | MR | Zbl

[TV06] Tao, Terence; Vu, Van Additive combinatorics, Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, 2006 | DOI | MR | Zbl

[TZ12] Tent, Katrin; Ziegler, Martin A course in model theory, Lecture Notes in Logic, 40, Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, 2012 | DOI | MR | Zbl

[Wei84] Weispfenning, Volker Quantifier elimination and decision procedures for valued fields, Models and sets (Aachen, 1983) (Lecture Notes in Mathematics), Volume 1103, Springer, 1984, pp. 419-472 | DOI | MR | Zbl