On the convex hull of integer points above the hyperbola

TL;DR

This paper analyzes the number of vertices in the convex hull of integer points above hyperbolas and provides bounds and an efficient enumeration algorithm, with implications for integer factorization.

Contribution

It establishes bounds on the vertices of convex hulls for points above hyperbolas and introduces a logarithmic-time vertex enumeration algorithm.

Findings

Vertices are between Ω(n^{1/3}) and O(n^{1/3} log n)

Bounds extend to hyperbolas with rational slopes

Algorithm enables efficient vertex enumeration

Abstract

We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola $\left\{xy = n\right\}$ has between $\Omega(n^{1/3})$ and $O(n^{1/3} \log n)$ vertices. The same bounds apply to any hyperbola with rational slopes except that instead of $n$ we have $n/\Delta$ in the lower bound and by $\max\left\{\Delta, n/\Delta\right\}$ in the upper bound, where $\Delta \in \mathbb{Z}_{>0}$ is the discriminant. We also give an algorithm that enumerates the vertices of these convex hulls in logarithmic time per vertex. One motivation for such an algorithm is the deterministic factorization of integers.