关于模双曲线上点的凸包
On the Convex Hull of the Points on Modular Hyperbolas
摘要：
给定整数$a$和$m\ge 2$，设$\Hm$为以下整数点集$$ \Hm= \{(x,y) \ : \ xy \equiv a \pmod m,\ 1\le x,y \le m-1\} $$我们改进了$v_a(m)$的几个已知上界，即$\Hm$的凸闭包的顶点数，并证明了在所有具有$\gcd(a,m)=1$的$a$上一致地存在$v_a(m) \le m^{1/2 + o(1)}$，而且对于几乎无平方的$m$，我们存在$v_a(m) \le m^{5/12 + o(1)}$.
英文摘要：
Given integers $a$ and $m\ge 2$, let $\Hm$ be the following set of integral points $$ \Hm= \{(x,y) \ : \ xy \equiv a \pmod m,\ 1\le x,y \le m-1\} $$ We improve several previously known upper bounds on $v_a(m)$, the number of vertices of the convex closure of $\Hm$, and show that uniformly over all $a$ with $\gcd(a,m)=1$ we have $v_a(m) \le m^{1/2 + o(1)}$ and furthermore, we have $v_a(m) \le m^{5/12 + o(1)}$ for $m$ which are almost squarefree.
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。
