On the Convex Hull of the Points on Modular Hyperbolas

Sergei V. Konyagin; Igor E. Shparlinski

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arXiv · 1012.1444

On the Convex Hull of the Points on Modular Hyperbolas

Sergei V. Konyagin·Igor E. Shparlinski

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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.

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BibTeX RIS

Sergei V. Konyagin, Igor E. Shparlinski. 2010-12-07. On the Convex Hull of the Points on Modular Hyperbolas. https://arxiv.org/abs/1012.1444

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