{
  "abstract": "We prove Szemerédi-Trotter-type incidence bounds for algebraic relations in any field $F$ admitting a valuation with finite residue field. This includes $\\mathbb F_q(t)$, $\\mathbb F_q((t))$, and all finitely generated extensions of finite fields. Bays and Martin show that any algebraic $K_{d,s}$-free relation $E\\subseteq F^k\\times F^\\ell$ 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 $A\\subseteq F^k$ and $B\\subseteq F^\\ell$, the number of edges $|E(A,B)|$ is at most $O(|A|^{1-ε}|B|^{1-\\frac{1-ε}{d}}+|A|+|B|)$. 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 $O(|P|^{\\frac{8}{9}}|L|^{\\frac{5}{9}}+|P|+|L|)$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37532",
  "author_total": 1,
  "authors": [
    {
      "name": "Mira Tartarotti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/33qmxt9qrwpfg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37532",
  "fulltext_url": null,
  "id": "33qmxt9qrwpfg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37532",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7224135436",
  "pdf_url": "https://arxiv.org/pdf/2610.37532",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/33qmxt9qrwpfg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37532",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37532"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Relative distality and incidence bounds in positive characteristic",
  "title_text": "Relative distality and incidence bounds in positive characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/33qmxt9qrwpfg",
  "venue": "arXiv",
  "versions": []
}