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Annals of Combinatorics

, Volume 23, Issue 2, pp 347–365 | Cite as

On the Relationship Between Ehrhart Unimodality and Ehrhart Positivity

  • Fu LiuEmail author
  • Liam Solus
Article
  • 7 Downloads

Abstract

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 \({\varvec{s}}\)-lecture hall simplices.

Keywords

Ehrhart positivity Ehrhart unimodality Real-rooted polynomial Lattice polytope Weighted projective space \({\varvec{s}}\)-lecture hall simplex Empty simplex Smooth polytope 

Mathematics Subject Classification

52B20 05A20 05A15 

Notes

Acknowledgements

Fu Liu was partially supported by a Grant from the Simons Foundation #426756. She was also supported by the National Science Foundation under Grant no. DMS-1440140 while she was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. Liam Solus was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship (DMS-1606407). The authors would like to thank Alexander Postnikov for posing the motivating question of this note. This research began at the 2017 MSRI Introductory Workshop to the Semester on Geometric and Topological Combinatorics and was continued at the 2017 Mini-Workshop on Lattice Polytopes at Oberwolfach. The authors are grateful to both institutions as well as the organizers of each event.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA
  2. 2.Matematik, KTHStockholmSweden

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