A $k$-Dimensional Version of the Largest Intersection Problem

Abstract

Suppose we have $r$ linearly independent hypersurfaces of degree $d$ in ${\mathbb{P}}^m$ (or ${\mathbb{A}}^m$) defined over a finite field ${\mathbb{F}}_q$, whose intersection is at most $k$-dimensional. What is the largest possible number of ${\mathbb{F}}_q$-rational points in the intersection? We conjecture an exact formula for this problem in both the projective and affine settings, assuming $q\ge d+1$. The case $k=m-1$ recovers the Beelen-Datta-Ghorpade conjecture [2] and the case $k=0$ recovers the zero-dimensional conjecture in [15]. Another interesting special case of the conjecture is $k=m-r$, which corresponds to the complete intersection of $r$ degree $d$ polynomials. The case $r=1$, $k=m-1$ was proven by Serre in [16] who showed that if $F$ is a degree $d$ homogeneous polynomial, then $|V(F)({\mathbb{F}}_q)|\le d{q}^{m-1}+{\pi }_{m-2}(q)$. We prove the case $r=2$ and $k=m-2$, that is, if $F_1,F_2$ are coprime, degree $d$ homogeneous polynomials, then $|V(F_1,F_2)({\mathbb{F}}_q)|\le d^2{q}^{m-2}+{\pi }_{m-3}(q)$.