Incidence theorems for multivariate polynomials over finite fields

Abstract

We study incidence problems for multivariate polynomials over a finite field ${\mathbb{F}}_q$. Given two families of $m$-variate polynomials, we count the number of triples $(f,g,x)$ such that $f$ belongs to the first family, $g$ belongs to the second family, $x\in {\mathbb{F}}_q^m$, and $f(x)=g(x)$. We show that for any subsets $\mathcal{L},{\mathcal{L}}^{\prime }\subseteq V_{m,r}$, where $V_{m,r}$ denotes the vector space of all $m$-variate polynomials over ${\mathbb{F}}_q$ of degree at most $r$, the number of such triples is at most $$q^{m-1}|\mathcal{L}||{\mathcal{L}}^{\prime }|+O\left(q^{\dim V_{m,r}-1}\sqrt{|\mathcal{L}||{\mathcal{L}}^{\prime }|}\right).$$ We further show that if $\mathcal{L}$ and ${\mathcal{L}}^{\prime }$ are contained in a subspace $V\subseteq V_{m,r}$ satisfying a suitable separating condition, then the same estimate holds with $\dim V_{m,r}$ replaced by $\dim V$. Our upper bound is essentially sharp when $q^{m-1}|\mathcal{L}||{\mathcal{L}}^{\prime }|$ dominates the summation. As applications, we derive incidence bounds for points and multivariate polynomials. These results recover and strengthen several previously known bounds for point-line incidences and point-univariate-polynomial incidences. Our proof is spectral, relying on an expander mixing lemma for general abelian Cayley color graphs together with Fourier analysis over finite fields.