Explicit Szemerédi–Trotter bounds over arbitrary fields
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.
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