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On the convex hull of integer points above the hyperbola

David Alcántara, Mónica Blanco, Francisco Criado, Francisco Santos

TL;DR

This work analyzes the convex hull of lattice points above a hyperbola, proving tight vertex bounds of $\Omega(n^{1/3})$ and $O(n^{1/3}\log n)$ for the standard hyperbola and extending to general hyperbolas with rational asymptotes via affine reductions. The authors derive an explicit upper bound using a rectangle cover and Andrews' lattice-polytope bound, and a matching lower bound by exploiting lattice lines parallel to the asymptotes. They then present a robust, $O(\log n)$-time-per-vertex algorithm to enumerate all vertices of the convex hull, including preprocessing to reduce to the standard hyperbola and a framework of minimum-slope ray casts and right-search-basis updates. The results yield a deterministic approach to factorization via lattice geometry and provide practical vertex-enumeration techniques for generalized hyperbolas. Overall, the paper combines geometric, combinatorial, and algorithmic methods to characterize and compute lattice convex hulls tied to hyperbolas with rational slopes.

Abstract

We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola $\left\{xy = n\right\}$ has between $Ω(n^{1/3})$ 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 $n/Δ$ in the lower bound and by $\max\left\{Δ, n/Δ\right\}$ in the upper bound, where $Δ\in \mathbb{Z}_{>0}$ 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.

On the convex hull of integer points above the hyperbola

TL;DR

This work analyzes the convex hull of lattice points above a hyperbola, proving tight vertex bounds of and for the standard hyperbola and extending to general hyperbolas with rational asymptotes via affine reductions. The authors derive an explicit upper bound using a rectangle cover and Andrews' lattice-polytope bound, and a matching lower bound by exploiting lattice lines parallel to the asymptotes. They then present a robust, -time-per-vertex algorithm to enumerate all vertices of the convex hull, including preprocessing to reduce to the standard hyperbola and a framework of minimum-slope ray casts and right-search-basis updates. The results yield a deterministic approach to factorization via lattice geometry and provide practical vertex-enumeration techniques for generalized hyperbolas. Overall, the paper combines geometric, combinatorial, and algorithmic methods to characterize and compute lattice convex hulls tied to hyperbolas with rational slopes.

Abstract

We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola has between and vertices. The same bounds apply to any hyperbola with rational slopes except that instead of we have in the lower bound and by in the upper bound, where 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.

Paper Structure

This paper contains 17 sections, 32 theorems, 84 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The number of vertices
  3. Reduction to the standard hyperbola
  4. Upper bound
  5. Lower Bound
  6. Experimental Results
  7. Algorithm to enumerate the vertices
  8. Preprocessing: Reduction to the standard hyperbola and computation of standard basis
  9. The algorithms
  10. Finding the first vertex
  11. Finding the next vertex
  12. Computing all vertices
  13. Starting at an intermediate point
  14. Proof of Theorem \ref{['thm:exists_find_nextpt']}
  15. Casting rays at the hyperbola
  16. ...and 2 more sections

Key Result

Theorem 1.1

There is a constant $C \in \mathbb{R}$ such that the following holds. If $h$ is an arbitrary hyperbola with asymptotes of rational slope, given by equation eq:hyp-intro with $a, b, c, \Delta \in \mathbb{Z}$, and $H$ is the closure of one of the convex components of $\mathbb{R}^2 \setminus h$, then where $m \coloneqq \max\{2\Delta, \frac{|n|}{\Delta}\}$.

Figures (6)

  • Figure 1: Rectangles $R_i$ and $U_i$ for $k=2$, $n = 14$ and $\Lambda = \mathbb{Z}^2$. By Lemma \ref{['lemma:upper-bound:rectangles']} this gives $i \in \{0,1,2,3\}$.
  • Figure 2: Number of vertices of $\text{\normalfontconv} \left({H_{n,\mathbb{Z}^2}}\right)$, plotted against the upper bound of $8.2 n^{1/3}(\log_2{n} + 2)$ and the lower bound of $2n^{1/3} - 2$. Plot is logarithmic in both axes.
  • Figure 3: Number of vertices of $\text{\normalfontconv} \left({H_{n,\mathbb{Z}^2}}\right)$ divided by $8.2n^{1/3} \log_2{n}$.
  • Figure 4: Number of vertices of $\text{\normalfontconv} \left({H_{n,\mathbb{Z}^2}}\right)$ divided by $2n^{1/3}$.
  • Figure 5: Case where $\mathop{\mathrm{nextpt}}\nolimits_{\mathbb{Z}^2,H_n}$ is not the next vertex.
  • ...and 1 more figures

Theorems & Definitions (64)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 1.7: Corollary \ref{['cor:find_next_vertex_from_non_vertex_log']}
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • ...and 54 more