A planar algebraic Zarankiewicz theorem over prime fields

Abstract

We prove an incidence bound for bipartite graphs on finite subsets of 𝔽²× 𝔽² defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is K_k,k-free and its vertex classes have sizes m and n, then it has O_t,k((mn)^2/3+m+n+mn/p) edges, where t bounds the description complexity, p is the characteristic of 𝔽, and 1/p=0 in characteristic zero. We also prove this bound for incidences between points and distinct geometrically irreducible components of a two-parameter polynomial family, allowing singular and nonreduced members. The proof extends Lewko's interpolation and contact-multiplicity method from lines to algebraic families. Applications include rich components, polynomial values on difference sets, and polynomial expansion.

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BibTeX RIS

BibTeX

@misc{quangham2026planar,
  title = {{A planar algebraic Zarankiewicz theorem over prime fields}},
  author = {Le Quang-Ham},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33327},
  eprint = {2609.33327},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33327},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A planar algebraic Zarankiewicz theorem over prime fields
AU  - Le Quang-Ham
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33327
UR  - https://arxiv.org/abs/2609.33327
AB  - We prove an incidence bound for bipartite graphs on finite subsets of 𝔽^2× 𝔽^2 defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is K_k,k-free and its vertex classes have sizes m and n, then it has O_t,k((mn)^2/3+m+n+mn/p) edges, where t bounds the description complexity, p is the characteristic of 𝔽, and 1/p=0 in characteristic zero. We also prove this bound for incidences between points and distinct geometrically irreducible components of a two-parameter polynomial family, allowing singular and nonreduced members. The proof extends Lewko's interpolation and contact-multiplicity method from lines to algebraic families. Applications include rich components, polynomial values on difference sets, and polynomial expansion.
ER  -

Abstract

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