On the Convex Hull of the Points on Modular Hyperbolas

TL;DR

This paper improves bounds on the number of vertices of the convex hull of points on modular hyperbolas, showing it grows slower than previously known, especially for almost squarefree moduli.

Contribution

It provides new upper bounds on the vertices of the convex hull of points on modular hyperbolas, refining previous estimates and applying to a broad class of moduli.

Findings

For all coprime a and m, v_a(m) ≤ m^{1/2 + o(1)}

For almost squarefree m, v_a(m) ≤ m^{5/12 + o(1)}

Improves previous bounds on the convex hull vertices of modular hyperbolas.

Abstract

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.