{"id":"palomar/PALOMAR-2026-10-07-000004","url":"https://github.com/mrklwk/szemeredi-trotter-lean/tree/844228bc6c042e59373888d1b69f4e215d1c00e4","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-07-000004-v1.json","repo":"https://github.com/mrklwk/szemeredi-trotter-lean","tags":["math.CO","msc:52C10","msc:05B25","incidence-geometry","combinatorics","finite-fields","lean-formalization"],"score":0,"title":"Explicit Szemerédi–Trotter bounds over arbitrary fields","trust":"high","domain":"math.CO","source":"palomar","authors":["Mark Lewko"],"reviews":0,"subject":"math.CO","summary":"We formalize four explicit Szemerédi–Trotter incidence bounds for m distinct points and n distinct affine lines in a field plane. In characteristic zero the bound is 3(mn)^(2/3) + m + n. In positive characteristic p, an arbitrary field admits the additional term 2mn/p. This term is absent when mn ≤ p³. Over the prime field ZMod p the additional term improves to mn/p without a size restriction. The development adapts the author's polynomial-method argument, uses generic coordinates for the improved linear term, and proves a finite-plane counting estimate for the prime-field refinement.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-07","reactions":{},"formalized":true,"formalizes":[{"title":"A Szemeredi-Trotter Theorem in Arbitrary Fields","identifier":"arXiv:2609.27023v2","relationship":"adapts"}]}