Relative distality and incidence bounds in positive characteristic
Abstract
We prove Szemerédi-Trotter-type incidence bounds for algebraic relations in any field F admitting a valuation with finite residue field. This includes 𝔽q(t), 𝔽q((t)), and all finitely generated extensions of finite fields. Bays and Martin show that any algebraic Kd,s-free relation E⊆ Fk× Fℓ admits a power-saving in the classical Zarankiewicz bound of Kővári, Sós, and Turán. Answering a question of the authors, we show that this power-saving is uniform in F: There is some ε>0 depending only on d and the complexity of E (and not on F) such that for A⊆ Fk and B⊆ Fℓ, the number of edges |E(A,B)| is at most O(|A|1−ε|B|1−(1−ε)/d+|A|+|B|). For the case where A is a set of points and B is a set of lines in F2 and E is the incidence relation, we prove the explicit bound O(|P|8/9|L|5/9+|P|+|L|). As the central ingredient of the proof, we introduce the model-theoretic notion of distality relative to a definable set. We begin developing this notion and prove that the theory of algebraically closed valued fields is distal relative to the residue field. An application of a result of Chernikov, Peterzil, and Starchenko, based on work of Chernikov, Galvin, and Starchenko, yields our incidence bounds.
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BibTeX
@misc{tartarotti2026relative,
title = {{Relative distality and incidence bounds in positive characteristic}},
author = {Mira Tartarotti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37532},
eprint = {2610.37532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37532},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Relative distality and incidence bounds in positive characteristic
AU - Mira Tartarotti
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37532
UR - https://arxiv.org/abs/2610.37532
AB - We prove Szemerédi-Trotter-type incidence bounds for algebraic relations in any field F admitting a valuation with finite residue field. This includes 𝔽_q(t), 𝔽_q((t)), and all finitely generated extensions of finite fields. Bays and Martin show that any algebraic K_d,s-free relation E⊆ F^k× F^ℓ admits a power-saving in the classical Zarankiewicz bound of Kővári, Sós, and Turán. Answering a question of the authors, we show that this power-saving is uniform in F: There is some ε>0 depending only on d and the complexity of E (and not on F) such that for A⊆ F^k and B⊆ F^ℓ, the number of edges |E(A,B)| is at most O(|A|^1−ε|B|^1−(1−ε)/d+|A|+|B|). For the case where A is a set of points and B is a set of lines in F^2 and E is the incidence relation, we prove the explicit bound O(|P|^8/9|L|^5/9+|P|+|L|). As the central ingredient of the proof, we introduce the model-theoretic notion of distality relative to a definable set. We begin developing this notion and prove that the theory of algebraically closed valued fields is distal relative to the residue field. An application of a result of Chernikov, Peterzil, and Starchenko, based on work of Chernikov, Galvin, and Starchenko, yields our incidence bounds.
ER -