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

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$forall$1\le i\le d$\mathrm{.}A$Q$-sphereisasetoftheform$\lbrace x\in $\mathbb{F}$_q^d | Q(x-b)=r\rbrace$,where$b\in $\mathbb{F}$_q^d, r\in $\mathbb{F}$_q$\mathrm{.}Weproveboundsonthenumberofincidencesbetweenapointset$$\mathcal{P}$$anda$Q$-sphereset$$\mathcal{S}$$,denotedby$I($\mathcal{P}$,$\mathcal{S}$)$,asthefollowing\mathrm{.}$$| I($\mathcal{P}$,$\mathcal{S}$)-\frac{|$\mathcal{P}$||$\mathcal{S}$|}{q}|\le q^{d/2}\sqrt{|$\mathcal{P}$||$\mathcal{S}$|}.$$ 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.