  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 $d$: $|X(\mathbb{F}_q)|\le dq^s+\pi_{s-1}$.
