On the convex hull of the points on multivariate modular hyperbolas

Igor E. Shparlinski

doi:10.1016/j.jnt.2016.07.011

On the convex hull of the points on multivariate modular hyperbolas

Igor E. Shparlinski

Research output: Contribution to journal › Article › peer-review

Abstract

Given integers a, m≥1 with gcd⁡(a,m)=1 and s≥2, let H_s(a,m) be the following set of integral pointsH_s(a,m)={(x₁,…,x_s)∈Z^s:x₁…x_s≡a(modm),={(x₁,…,x_s)∈Z^s:1≤x₁,…,x_s≤m−1}. We obtain upper bounds on the number of vertices of the convex hull of H_s(a,m). These bounds generalise those known for s=2, although our approach is different.

Original language	English
Pages (from-to)	71-78
Number of pages	8
Journal	Journal of Number Theory
Volume	171
DOIs	https://doi.org/10.1016/j.jnt.2016.07.011
Publication status	Published - 1 Feb 2017
Externally published	Yes

Keywords

Convex hull
Modular hyperbola

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10.1016/j.jnt.2016.07.011

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abstract = "Given integers a, m≥1 with gcd⁡(a,m)=1 and s≥2, let Hs(a,m) be the following set of integral pointsHs(a,m)=\{(x1,…,xs)∈Zs:x1…xs≡a(modm),=\{(x1,…,xs)∈Zs:1≤x1,…,xs≤m−1\}. We obtain upper bounds on the number of vertices of the convex hull of Hs(a,m). These bounds generalise those known for s=2, although our approach is different.",

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N2 - Given integers a, m≥1 with gcd⁡(a,m)=1 and s≥2, let Hs(a,m) be the following set of integral pointsHs(a,m)={(x1,…,xs)∈Zs:x1…xs≡a(modm),={(x1,…,xs)∈Zs:1≤x1,…,xs≤m−1}. We obtain upper bounds on the number of vertices of the convex hull of Hs(a,m). These bounds generalise those known for s=2, although our approach is different.

AB - Given integers a, m≥1 with gcd⁡(a,m)=1 and s≥2, let Hs(a,m) be the following set of integral pointsHs(a,m)={(x1,…,xs)∈Zs:x1…xs≡a(modm),={(x1,…,xs)∈Zs:1≤x1,…,xs≤m−1}. We obtain upper bounds on the number of vertices of the convex hull of Hs(a,m). These bounds generalise those known for s=2, although our approach is different.

KW - Convex hull

KW - Modular hyperbola

UR - https://www.scopus.com/pages/publications/84986245793

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DO - 10.1016/j.jnt.2016.07.011

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JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

On the convex hull of the points on multivariate modular hyperbolas

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Keywords

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