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PreprintCC BY-NC-SA 4.0arXiv

Two proofs and a tight Szemerédi--Trotter theorem

  • Hung-Hsun Hans Yu
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Abstract

The Szemerédi--Trotter theorem, a fundamental theorem in incidence geometry, states that n points and m lines in ℝ2 form at most O(m+n+m2/3n2/3) incidences. Recently, Lewko and independently Miao--Xie extended the Szemerédi--Trotter theorem to arbitrary fields with a correction term O(mn/p) if the field has positive characteristic 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 p, n≤ m and p2(k−1)≤ m<p2k for some positive integer k, the number of incidences can be bounded by O(pk−1n+m+m2/3n2/3+mn/pk).

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Cite

BibTeXRIS
BibTeX
@misc{yu2026proofs,
  title = {{Two proofs and a tight Szemerédi--Trotter theorem}},
  author = {Hung-Hsun Hans Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10649},
  eprint = {2610.10649},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10649},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Two proofs and a tight Szemerédi--Trotter theorem
AU  - Hung-Hsun Hans Yu
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10649
UR  - https://arxiv.org/abs/2610.10649
AB  - The Szemerédi--Trotter theorem, a fundamental theorem in incidence geometry, states that n points and m lines in ℝ^2 form at most O(m+n+m^2/3n^2/3) incidences. Recently, Lewko and independently Miao--Xie extended the Szemerédi--Trotter theorem to arbitrary fields with a correction term O(mn/p) if the field has positive characteristic 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 p, n≤ m and p^2(k−1)≤ m<p^2k for some positive integer k, the number of incidences can be bounded by O(p^k−1n+m+m^2/3n^2/3+mn/p^k).
ER  -