Ehrhart Theory of Spanning Lattice Polytopes

Hofscheier, Johannes; Katth�n, Lukas; Nill, Benjamin

Abstract

© 2018 Oxford University Press. All rights reserved. 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 article, we prove a new result on the vanishing of its coefficients. As a consequence, we get that hi = 0 implies hi+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.

Citation

Hofscheier, J., Katthän, L., & Nill, B. (2018). Ehrhart Theory of Spanning Lattice Polytopes. International Mathematics Research Notices, 2018(19), 5947-5973. https://doi.org/10.1093/imrn/rnx065

• Journal Article Type: Article
• Acceptance Date: Feb 27, 2017
• Online Publication Date: Mar 26, 2017
• Publication Date: Oct 9, 2018
• Deposit Date: Nov 12, 2019
• Publicly Available Date: Nov 14, 2019
• Journal: International Mathematics Research Notices
• Print ISSN: 1073-7928
• Electronic ISSN: 1687-0247
• Publisher: Oxford University Press
• Peer Reviewed: Peer Reviewed
• Volume: 2018
• Issue: 19
• Pages: 5947-5973
• DOI: https://doi.org/10.1093/imrn/rnx065
• Keywords: General Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/3234036
• Publisher URL: https://academic.oup.com/imrn/article/2018/19/5947/3091116
• Contract Date: Nov 14, 2019