On the Convex Hull of the Points on Modular Hyperbolas

Details On the Convex Hull of the Points on Modular Hyperbolas On the Convex Hull of the Points on Modular Hyperbolas

2010-12-07

arxiv.org/abs/1012.1444v2

Mathematics

Number Theory

Scientific paper

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.

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