On the relationship between Ehrhart unimodality and Ehrhart positivity. (English) Zbl 1418.52004

Liu, Fu; Solus, Liam

Ann. Comb. 23, No. 2, 347-365 (2019).

Summary: For a given lattice polytope, two fundamental problems within the field of Ehrhart theory are (1) to determine if its (Ehrhart) \(h^*\)-polynomial is unimodal and (2) to determine if its Ehrhart polynomial has only positive coefficients. The former property of a lattice polytope is known as Ehrhart unimodality and the latter property is known as Ehrhart positivity. These two properties are often simultaneously conjectured to hold for interesting families of lattice polytopes, yet they are typically studied in parallel. As to answer a question posed at the 2017 Introductory Workshop to the MSRI Semester on Geometric and Topological Combinatorics, the purpose of this note is to show that there is no general implication between these two properties in any dimension greater than two. To do so, we investigate these two properties for families of well-studied lattice polytopes, assessing one property where previously only the other had been considered. Consequently, new examples of each phenomena are developed, some of which provide an answer to an open problem in the literature. The well-studied families of lattice polytopes considered include zonotopes, matroid polytopes, simplices of weighted projective spaces, empty lattice simplices, smooth polytopes, and \(\boldsymbol{s}\)-lecture hall simplices.

MSC:

52B20	Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
05A20	Combinatorial inequalities
05A15	Exact enumeration problems, generating functions

Keywords:

Ehrhart positivity; Ehrhart unimodality; real-rooted polynomial; lattice polytope; weighted projective space; \(\boldsymbol{s}\)-lecture Hall simplex; empty simplex; smooth polytope

Software:

MathOverflow; polymake

References:

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