Incidences between quadratic subspaces over finite fields

TL;DR

This paper establishes bounds on the incidences between specific quadratic subspaces over finite fields, improving previous error estimates under certain conditions.

Contribution

It introduces new bounds for incidences between dot_k and dot_h quadratic subspaces over finite fields, refining earlier results by Phuong, Thang, and Vinh.

Findings

Derived bounds for incidences between quadratic subspaces

Improved error terms over previous results

Applicable to collections of affine subspaces with additional conditions

Abstract

Let $\mathbb{F}_{q}$ be a finite field of order $q$, where $q$ is an odd prime power. A quadratic subspace $(W,Q)$ of $(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})$ is called dot$_{k}$-subspace if $Q$ is isometrically isomorphic to $x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}$. In this paper, we obtain bounds for the number of incidences $I(\mathcal{K},\mathcal{H})$ between a collection $\mathcal{K}$ of dot$_{k}$-subspaces and a collection $\mathcal{H}$ of dot$_{h}$-subspaces when $h \geq 4k-4$, which is given by \[\left | I(\mathcal{K},\mathcal{H})-\frac{|\mathcal{K}||\mathcal{H}|}{q^{k(n-h)}}\right | \lesssim q^{\frac{k(2h-n-2k+4)+h(n-h-1)-2}{2}}\sqrt{|\mathcal{K}||\mathcal{H}|}. \] In particular, we improve the error term obtained by Phuong, Thang and Vinh (2019) for general collections of affine subspaces in the presence of our additional conditions.