Lattice-point generating functions for free sums of convex sets

Beck, Matthias; Jayawant, Pallavi; McAllister, Tyrrell B.

doi:10.1016/j.jcta.2013.03.007

Cited by 18 publications

(22 citation statements)

References 14 publications

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“…, e n−1 , v 0 ) is a sum of a unique element from L 1 and a unique element from L 2 . Together, these facts mean F 0 is the affine free sum of G 1 and G 2 , as introduced in [3]. Since G 2 is a standard simplex, its normalized volume is 1; moreover, it is known that the k-dimensional standard simplex ∆ k is Gorenstein of index k + 1, that is, there exists a unique vector v ∈ Z k (namely, v = (−1, .…”

Section: Cycle Graphsmentioning

confidence: 99%

Counting Equilibria of the Kuramoto Model Using Birationally Invariant Intersection Index

Chen¹,

Davis²,

Mehta³

2018

SIAM J. Appl. Algebra Geometry

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Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appear naturally in many independent fields of science and engineering. A substantial amount of work has been devoted to understanding all possible synchronization configurations on a given network. In this setting, a key problem is to determine the total number of such configurations. Through an algebraic formulation, for tree and cycle graphs, we provide an upper bound on this number using the birationally invariant intersection index of a system of rational functions on a toric variety.

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Section: Cycle Graphsmentioning

confidence: 99%

Counting Equilibria of the Kuramoto Model Using Birationally Invariant Intersection Index

Chen¹,

Davis²,

Mehta³

2018

SIAM J. Appl. Algebra Geometry

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“…It is straightforward to compute that the lattice points in ∆ (1,q) are the columns of the following matrix.  (1,2,2,15,20,20) .…”

Section: The Integer Decomposition Property and ∆mentioning

confidence: 99%

“…It is straightforward to verify that this point is not the sum of exactly two lattice points in ∆ (1,2,2,15,20,20) , and thus this simplex is not IDP. However, (2, 2, 15, 20, 20) satisfies our linear system.…”

Section: The Integer Decomposition Property and ∆mentioning

confidence: 99%

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Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices

Braun

Davis

Solus

2018

Advances in Applied Mathematics

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A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart) h * -polynomial. This conjecture can be viewed as a strengthening of a previously disproved conjecture which stated that any Gorenstein lattice polytope has a unimodal h * -polynomial. The first counterexamples to unimodality for Gorenstein lattice polytopes were given in even dimensions greater than five by Mustaţǎ and Payne, and this was extended to all dimensions greater than five by Payne. While there exist numerous examples in support of the conjecture that IDP reflexives are h * -unimodal, its validity has not yet been considered for families of reflexive lattice simplices that closely generalize Payne's counterexamples. The main purpose of this work is to prove that the former conjecture does indeed hold for a natural generalization of Payne's examples. The second purpose of this work is to extend this investigation to a broader class of lattice simplices, for which we present new results and open problems.

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“…The conditions needed for Braun's formula to hold were significantly relaxed in [3], including a multivariate generalization. However, we will not need such power here.…”

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confidence: 99%

A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

Chen

Davis

2019

Springer Proceedings in Mathematics &Amp; Statistics

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The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands. IntroductionA (convex) polytope P ⊂ R n is the convex hull of finitely many points in R n . Equivalently, P is the bounded intersection of finitely many closed half-spaces in R n . We call a polytope lattice if its vertices are elements of Z n . Polytopes are fascinating combinatorial objects, and lattice polytopes are especially interesting due to their appearance in many contexts, such as combinatorics, algebraic statistics, and physics; see, for example, [2,26,27] Given polytopes P ⊂ R m and Q ⊂ R n , set P ⊕ Q := conv{(P, 0) ∪ (0, Q)} ⊂ R m+n .If P and Q each contain the origin, then we call P ⊕ Q the free sum of P and Q. It will be convenient to use the notation P ′ , Q ′ for (P, 0), (0, Q) ∈ R m+n , respectively, so that we simply write P ⊕ Q = conv{P ′ ∪ Q ′ }. In the present contribution, we

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Lattice-point generating functions for free sums of convex sets

Cited by 18 publications

References 14 publications

Counting Equilibria of the Kuramoto Model Using Birationally Invariant Intersection Index

Counting Equilibria of the Kuramoto Model Using Birationally Invariant Intersection Index

Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices

A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

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