Lattice polytopes with distinct pair-sums

M. D. Choi; T. Y. Lam; B. Reznick

doi:10.1007/s00454-001-0052-9

Lattice polytopes with distinct pair-sums

M. D. Choi, T. Y. Lam, B. Reznick

Mathematics

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

Abstract

Let P be a lattice polytope in Rⁿ, and let P ∩ Zⁿ = {v₁,..., v_N}. If the N+ (₂^N) points 2v₁,..., 2v_N; v₁+v₂,..., v_N-1 + v_N are distinct, we say that P is a "distinct pair-sum" or "dps" polytope. We show that if P is a dps polytope in Rⁿ, then N ≤2ⁿ, and, for every n, we construct dps polytopes in Rⁿ which contain 2ⁿ lattice points. We also discuss the relation between dps polytopes and the study of sums of squares of real polynomials.

Original language	English (US)
Pages (from-to)	65-72
Number of pages	8
Journal	Discrete and Computational Geometry
Volume	27
Issue number	1
DOIs	https://doi.org/10.1007/s00454-001-0052-9
State	Published - Jan 2002

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1007/s00454-001-0052-9

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N2 - Let P be a lattice polytope in Rn, and let P ∩ Zn = {v1,..., vN}. If the N+ (2N) points 2v1,..., 2vN; v1+v2,..., vN-1 + vN are distinct, we say that P is a "distinct pair-sum" or "dps" polytope. We show that if P is a dps polytope in Rn, then N ≤2n, and, for every n, we construct dps polytopes in Rn which contain 2n lattice points. We also discuss the relation between dps polytopes and the study of sums of squares of real polynomials.

AB - Let P be a lattice polytope in Rn, and let P ∩ Zn = {v1,..., vN}. If the N+ (2N) points 2v1,..., 2vN; v1+v2,..., vN-1 + vN are distinct, we say that P is a "distinct pair-sum" or "dps" polytope. We show that if P is a dps polytope in Rn, then N ≤2n, and, for every n, we construct dps polytopes in Rn which contain 2n lattice points. We also discuss the relation between dps polytopes and the study of sums of squares of real polynomials.

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Lattice polytopes with distinct pair-sums

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