The degree of point configurations: Ehrhart theory, Tverberg points and almost neighborly polytopes

Nill, Benjamin; Padrol, Arnau

doi:10.1016/j.ejc.2015.03.024

Cited by 5 publications

(3 citation statements)

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“…Let us briefly discuss the relation of the results of this section to the study of point configurations of small combinatorial degree, i. e., the maximal degree of the h-vector of lattice triangulations of P . We refer to [NP15] for terminology and background. Let us observe that the h-vector of a lattice triangulation T of P has no internal zeros.…”

Section: Corollary 31 There Are Only Finitely Many Lattice Polytopementioning

confidence: 99%

Ehrhart Theory of Spanning Lattice Polytopes

Hofscheier

Katthän

Nill

2017

International Mathematics Research Notices

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The key object in the Ehrhart theory of lattice polytopes is the numerator polynomial of the rational generating series of the Ehrhart polynomial, called h * -polynomial. In this paper we prove a new result on the vanishing of its coefficients. As a consequence, we get that h * i = 0 implies h * i+1 = 0 if the lattice points of the lattice polytope affinely span the ambient lattice. This generalizes a recent result in algebraic geometry due to Blekherman, Smith, and Velasco, and implies a polyhedral consequence of the Eisenbud-Goto conjecture. We also discuss how this study is motivated by unimodality questions and how it relates to decomposition results on lattice polytopes of given degree. The proof methods involve a novel combination of successive modifications of half-open triangulations and considerations of number-theoretic step functions.

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Section: Corollary 31 There Are Only Finitely Many Lattice Polytopementioning

confidence: 99%

Ehrhart Theory of Spanning Lattice Polytopes

Hofscheier

Katthän

Nill

2017

International Mathematics Research Notices

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“…. , P n ) = deg(P ).This unmixed situation has been studied rather intensively (e.g., [4,21,16]) leading to applications and relations to the adjunction theory of polarized toric varieties [12,11,1], dual defective toric varieties [13], and almost-neighborly point configurations [22]. We hope to eventually generalize some of the achieved results to the mixed situation.…”

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confidence: 93%

The Mixed Degree of Families of Lattice Polytopes

Nill

2020

Ann. Comb.

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The degree of a lattice polytope is a notion in Ehrhart theory that was studied quite intensively over the previous years. It is well-known that a lattice polytope has normalized volume one if and only if its degree is zero. Recently, Esterov and Gusev gave a complete classification result of families of n lattice polytopes in R n whose mixed volume equals one. Here, we give a reformulation of their result involving the novel notion of a mixed degree that generalizes the degree similar to how the mixed volume generalizes the volume. We discuss and motivate this terminology, and explain why it extends a previous definition of Soprunov. We also remark how a recent combinatorial result due to Bihan solves a related problem posed by Soprunov. Definitions and motivation1.1. Introduction. Lattice polytopes in R n are called hollow (or lattice-free) [23,8] if they have no lattice points (i.e., elements in Z n ) in their relative interiors. In this paper, we initiate the study of large families of lattice polytopes with hollow Minkowski sums. We observe that such a family can consist of at most n elements (Proposition 2.1). In Theorem 2.2, we deduce from the main result in [14] that a family of n lattice polytopes in R n has mixed volume one if and only if the Minkowski sums of all subfamilies are hollow. In order to measure the 'hollowness' of a family of lattice polytopes, we introduce the mixed degree of a family of lattice polytopes. Our goal is to convince the reader that this is a worthwhile to study invariant of a family of lattice polytopes that naturally generalizes the much-studied notion of the degree of a lattice polytope in a manner similar to how the mixed volume generalizes the normalized volume (see Subsection 1.3). As first positive evidence for this claim, we show the nonnegativity of the mixed degree (Subsection 2.1), a generalization of the nonnegativity of the degree, and the characterization of mixed degree zero by mixed volume one (Subsection 2.2) in analogy to the characterization of degree zero by normalized volume one. We will also explain how the definition given here generalizes an independent definition of Soprunov (Subsection 2.3). Basic definitions.Let us recall that a lattice polytope P ⊂ R n is a polytope whose vertices are elements of the lattice Z n . Two lattice polytopes are unimodularly equivalent if they are isomorphic via an affine lattice-preserving transformation. We denote by conv(A) the convex hull of a set A ⊆ R n . We say P is an n-dimensional unimodular simplex if it is unimodularly equivalent to ∆ n := conv(0, e 1 , . . . , e n ), where 0 denotes the origin of R n and e 1 , . . . , e n the standard basis vectors. We define the normalized volume Vol(P ) as dim(P )! times the Euclidean volume with respect to the affine lattice given by the intersection of Z n and the affine span of P . Note that Vol(∆ n ) = 1.Definition 1.1. Let P 1 , . . . , P m ⊂ R n be a finite set of lattice polytopes.• For k ∈ Z ≥1 we set [k] := {1, . . . , k}.• For ∅ = I ⊆ [m] we define their Minkowski ...

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“…Sets satisfying property (iii) of Lemma 11 are called of combinatorial degree one by B. Nill, A. Padrol [4], who give a complete classification of them. The description uses iterated pyramids, which we define in terms of the join operator.…”

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confidence: 99%