On the convex hull of integer points above the hyperbola

2025-01-31

AI Summary

This work investigates the asymptotic behavior of the number of vertices of the convex hull of integer lattice points lying above the hyperbola $xy = n$, and addresses the efficient enumeration of these vertices. Methodologically, it integrates tools from integer lattice geometry, convex hull theory, continued fractions, and Diophantine approximation; leveraging the underlying lattice structure, it designs a deterministic algorithm. The main contributions are: (i) establishing the first tight asymptotic bound $Theta(n^{1/3} log^{O(1)} n)$ on the number of vertices—significantly improving upon prior results where upper and lower bounds were separated; and (ii) extending this analysis to arbitrary rational-slope hyperbolas. Furthermore, it introduces the first deterministic vertex enumeration algorithm achieving $O(log n)$ time per vertex, circumventing the high overhead of generic convex hull algorithms. This work provides novel theoretical foundations and algorithmic support for geometry-based deterministic integer factorization.

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Abstract

We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola $left{xy = n ight}$ 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 $maxleft{Delta, n/Delta ight}$ 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.

Problem

• Convex Polygons
• Hyperbolic Curve
• Angle Count Adjustment

Innovation

• ConvexPolygonsHyperbola
• IntegerPointsComplexity
• EfficientFactorizationAlgorithm