{
  "abstract": "The Szemerédi--Trotter theorem, a fundamental theorem in incidence geometry, states that $n$ points and $m$ lines in $\\mathbb{R}^2$ form at most $O(m+n+m^{2/3}n^{2/3})$ incidences. Recently, Lewko and independently Miao--Xie extended the Szemerédi--Trotter theorem to arbitrary fields with a correction term $O(\\frac{mn}{\\mathfrak{p}})$ if the field has positive characteristic $\\mathfrak{p}$. In this paper, we provide some geometric intuition for the two proofs and compare them. Moreover, we show how both proofs can be improved using Hasse derivatives. As a consequence, we obtain a tight upper bound over any algebraically closed field. To be specific, when the field has positive characteristic $\\mathfrak{p}$, $n\\leq m$ and $\\mathfrak{p}^{2(k-1)}\\leq m<\\mathfrak{p}^{2k}$ for some positive integer $k$, the number of incidences can be bounded by \\[O\\left(\\mathfrak{p}^{k-1}n+m+m^{2/3}n^{2/3}+\\frac{mn}{\\mathfrak{p}^{k}}\\right).\\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10649",
  "author_total": 1,
  "authors": [
    {
      "name": "Hung-Hsun Hans Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/drqnevx7v5rpe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10649",
  "fulltext_url": null,
  "id": "drqnevx7v5rpe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10649",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.HO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7221092058",
  "pdf_url": "https://arxiv.org/pdf/2610.10649",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/drqnevx7v5rpe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10649",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10649"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Two proofs and a tight Szemerédi--Trotter theorem",
  "title_text": "Two proofs and a tight Szemerédi--Trotter theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/drqnevx7v5rpe",
  "venue": "arXiv",
  "versions": []
}