AI 中文总结
本文研究有限域上超曲面交集的有理点计数问题，提出k维版本最大相交问题的精确公式猜想，并证明了r=2、k=m-2的特殊情形。
AI 中文摘要
假设在有限域$\u2119_q$上定义了$\u2112^m$（射影空间）或$\u2112^m$（仿射空间）中r个次数为d的线性无关超曲面，它们的交集维数至多为k，该交集中$\u2119_q$-有理点的最大可能数量是多少？我们猜想，在射影和仿射两种情形下，当$q\uf8ff d+1$时，该问题存在精确公式。当$k=m-1$时，该猜想退化为Beelen-Datta-Ghorpade猜想[2]；当$k=0$时，退化为文献[15]中的零维猜想。该猜想的另一个有趣特例是$k=m-r$，对应r个次数为d的多项式的完全交集。Serre在文献[16]中证明了$r=1$、$k=m-1$的情形，即若F是次数为d的齐次多项式，则$|V(F)(\u2119_q)|\uf8ff dq^{m-1}+\u03c0_{m-2}(q)$。本文证明了$r=2$、$k=m-2$的情形，即若$F_1$、$F_2$是互素的次数为d的齐次多项式，则$|V(F_1,F_2)(\u2119_q)|\uf8ff d^2q^{m-2}+\u03c0_{m-3}(q)$。
英文摘要
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 $q\geq d+1$. 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 $|V(F)(\mathbb{F}_q)|\leq dq^{m-1}+π_{m-2}(q)$. We prove the case $r=2$ and $k=m-2$, that is, if $F_1, F_2$ are coprime, degree $d$ homogeneous polynomials, then $|V(F_1,F_2)(\mathbb{F}_q)|\leq d^2q^{m-2}+π_{m-3}(q)$.
