$\mathbb{P}^{n}(\mathbb{F}_q)$中两个无公共超平面的二次曲面的交
Intersection of two quadrics with no common hyperplane in $\mathbb{P}^{n}(\mathbb{F}_q)$}}
摘要：
设$\mathcal{Q}_1$和$\mathcal{Q}_2$是${\mathbb{P}}^n(\mathbb{F}_q)$中没有公共超平面的两个任意二次曲面。我们给出了这两个二次曲面相交点的个数的最佳上界。我们的结果表明，$| \mathcal{Q}_1\cap \mathcal{Q}_2|\le 4q^{n-2}+π_{n-3}$。这个结果启发我们建立了关于维数为$s$，次数为$d$的代数集$X\subset {\mathbb{P}}^n(\mathbb{F}_q)$的点数的猜想：$|X(\mathbb{F}_q)|\le dq^s+π_{s-1}$ .
英文摘要：
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 $| \mathcal{Q}_1\cap \mathcal{Q}_2|\le 4q^{n-2}+π_{n-3}$. 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 $d$: $|X(\mathbb{F}_q)|\le dq^s+π_{s-1}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
