On the convex hull of integer points above the hyperbola
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912213077327872 |
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| author | Alcántara, David Blanco, Mónica Criado, Francisco Santos, Francisco |
| author_facet | Alcántara, David Blanco, Mónica Criado, Francisco Santos, Francisco |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_19193 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the convex hull of integer points above the hyperbola Alcántara, David Blanco, Mónica Criado, Francisco Santos, Francisco Combinatorics Computational Geometry 52C05, 52C45, 68R99, 52-08 I.3.5 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. |
| title | On the convex hull of integer points above the hyperbola |
| topic | Combinatorics Computational Geometry 52C05, 52C45, 68R99, 52-08 I.3.5 |
| url | https://arxiv.org/abs/2501.19193 |