Let $\mathbb{ F}_q$ be a finite field of $q$ elements where $q$ is a large odd prime power and $Q =a_1 x_1^{ c_1}+...+a_dx_d^{ c_d}\in \mathbb{ F}_q[x_1,...,x_d]$, where $2\le c_i\le N$, $\gcd(c_i,q)=1$, and $a_i\in \mathbb{ F}_q$ for all $1\le i\le d$. A $Q$-sphere is a set of the form $\lbrace x\in \mathbb{ F}_q^d | Q(x-b)=r\rbrace$, where $b\in \mathbb{ F}_q^d, r\in \mathbb{ F}_q$. We prove bounds on the number of incidences between a point set $\mathcal{ P}$ and a $Q$-sphere set $\mathcal{ S}$, denoted by $I(\mathcal{ P},\mathcal{ S})$, as the following. $$| I(\mathcal{ P},\mathcal{ S})-\frac{ |\mathcal{ P}||\mathcal{ S}|}{ q}|\le q^{ d/2}\sqrt{ |\mathcal{ P}||\mathcal{ S}|}.$$ We prove this estimate by studying the spectra of directed graphs. We also give a version of this estimate over finite rings $\mathbb{ Z}_q$ where $q$ is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem. In Sections $4$ and $5$, we prove a bound on the number of incidences between a random point set and a random $Q$-sphere set in $\mathbb{ F}_q^d$. 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.
Categories: math.CO