Incidences between quadratic subspaces over finite fields

2020/05/25 by Semin Yoo, Yoo, Semin
Computer Science · Mathematics · #Coding theory and cryptography #Finite Group Theory Research #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2005.12255

Abstract

Let \mathbbF_q be a finite field of order q, where q is an odd prime power. A quadratic subspace (W,Q) of (\mathbbF_qⁿ,x₁²+x₂²+⋯+x_n²) is called dot_k-subspace if Q is isometrically isomorphic to x₁²+x₂²+⋯+x_k². In this paper, we obtain bounds for the number of incidences I(K,H) between a collection K of dot_k-subspaces and a collection H of dot_h-subspaces when h ≥ 4k-4, which is given by | I(K,H)-\frac|K||H|q^k(n-h) | \lesssim q^{(k(2h-n-2k+4)+h(n-h-1)-2)/(2)}√(|K||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.

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