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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

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Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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1435-5337
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Volume 29, Issue 2

# Incidences between points and generalized spheres over finite fields and related problems

Nguyen D. Phuong
/ Pham Thang
/ Le A. Vinh
Published Online: 2016-06-02 | DOI: https://doi.org/10.1515/forum-2015-0024

## Abstract

Let ${𝔽}_{q}$ be a finite field of q elements, where q is a large odd prime power and

$Q={a}_{1}{x}_{1}^{{c}_{1}}+\mathrm{\cdots }+{a}_{d}{x}_{d}^{{c}_{d}}\in {𝔽}_{q}\left[{x}_{1},\mathrm{\dots },{x}_{d}\right],$

where $2\le {c}_{i}\le N$, $\mathrm{gcd}\left({c}_{i},q\right)=1$, and ${a}_{i}\in {𝔽}_{q}$ for all $1\le i\le d$. A Q -sphere is a set of the form

$\left\{𝒙\in {𝔽}_{q}^{d}\mid Q\left(𝒙-𝒃\right)=r\right\},$

where $𝒃\in {𝔽}_{q}^{d},r\in {𝔽}_{q}$. We prove bounds on the number of incidences between a point set $\mathcal{𝒫}$ and a Q-sphere set $\mathcal{𝒮}$, denoted by $I\left(\mathcal{𝒫},\mathcal{𝒮}\right)$, as the following:

$|I\left(\mathcal{𝒫},\mathcal{𝒮}\right)-\frac{|\mathcal{𝒫}||\mathcal{𝒮}|}{q}|\le {q}^{d/2}\sqrt{|\mathcal{𝒫}||\mathcal{𝒮}|}.$

We also give a version of this estimate over finite cyclic rings $ℤ/qℤ$, where q is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem and a bound on the number of incidences between a random point set and a random Q-sphere set in ${𝔽}_{q}^{d}$. We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.

MSC 2010: 52C10

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Revised: 2016-01-26

Published Online: 2016-06-02

Published in Print: 2017-03-01

The first and third author were supported by Vietnam National Foundation for Science and Technology grant 101.99-2013.21. The work of the second author was partially supported by Swiss National Science Foundation grants 200020-144531 and 200021-137574.

Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 449–456, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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