Intersection of two quadrics with no common hyperplane in ${\mathbb{P}}^n({\mathbb{F}}_q)$}}

Abstract

Let ${\mathcal{Q}}_1$ and ${\mathcal{Q}}_2$ be two arbitrary quadrics with no common hyperplane in ${\mathbb{P}}^n({\mathbb{F}}_q)$. We give the best upper bound for the number of points in the intersection of these two quadrics. Our result states that $|{\mathcal{Q}}_1\cap {\mathcal{Q}}_2|\le 4q^{n-2}+{\pi }_{n-3}$. This result inspires us to establish the conjecture on the number of points of an algebraic set $X\subset {\mathbb{P}}^n({\mathbb{F}}_q)$ of dimension $s$ and degree $d$: $|X({\mathbb{F}}_q)|\le d{q}^s+{\pi }_{s-1}$.

8 pages