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arXiv · 2408.10977

A point-variety incidence theorem over finite fields, and its applications

Xiangliang Kong·Itzhak Tamo
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Abstract

Incidence problems between geometric objects is a key area of focus in the field of discrete geometry. Among them, the study of incidence problems over finite fields have received a considerable amount of attention in recent years. In this paper, by characterizing the singular values and singular vectors of the corresponding incidence matrix through group algebras, we prove a bound on the number of incidences between points and varieties of a certain form over finite fields. Our result leads to a new incidence bound for points and flats in finite geometries, which improves previous results for certain parameter regimes. As another application of our point-variety incidence bound, we extend a result on pinned distance problems by Phuong, Thang, and Vinh, and independently by Cilleruelo, Iosevich, Lund, Roche-Newton, and Rudnev, under a weaker condition.

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Xiangliang Kong, Itzhak Tamo. 2024-08-20. A point-variety incidence theorem over finite fields, and its applications. https://arxiv.org/abs/2408.10977

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