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Examples of IDP lattice polytopes with non-log-concave $h^*$-vector

Johannes Hofscheier, Vadym Kurylenko, Benjamin Nill

TL;DR

The paper addresses whether the $h^*$-vector of IDP lattice polytopes is unimodal and, more strictly, log-concave, and reports counterexamples to log-concavity. It constructs explicit counterexamples and uses arc polytopes from directed graphs; the search benefited from reinforcement-learning-inspired methods and software verification. Contributions include a $7$-dimensional IDP polytope with $h^*$-vector $\bigl(1,2,3,4,5,3,2,1\bigr)$ and a $12$-dimensional unimodular $0/1$-polytope with $h^*$-vector $\bigl(1,2,3,4,5,3,2,1,0,0,0,0,0\bigr)$, both with positive Ehrhart coefficients $E_P(t)$; the latter corresponds to the arc polytope of a directed bipartite graph. These results show that log-concavity need not hold for IDP polytopes, informing the search for precise conditions under which $h^*$-vectors are unimodal or log-concave; additional details and more examples will be provided in future work.

Abstract

Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i.e., any lattice point in a $k$th dilation is a sum of $k$ lattice points in the polytope. It is a long-standing conjecture whether the numerator of the Ehrhart series of an IDP polytope, called the $h^*$-polynomial, has a unimodal coefficient vector. In this preliminary report on research in progress we present examples showing that $h^*$-vectors of IDP polytopes do not have to be log-concave. This answers a question of Luis Ferroni and Akihiro Higashitani. As this is an ongoing project, this paper will be updated with more details and examples in the near future.

Examples of IDP lattice polytopes with non-log-concave $h^*$-vector

TL;DR

The paper addresses whether the -vector of IDP lattice polytopes is unimodal and, more strictly, log-concave, and reports counterexamples to log-concavity. It constructs explicit counterexamples and uses arc polytopes from directed graphs; the search benefited from reinforcement-learning-inspired methods and software verification. Contributions include a -dimensional IDP polytope with -vector and a -dimensional unimodular -polytope with -vector , both with positive Ehrhart coefficients ; the latter corresponds to the arc polytope of a directed bipartite graph. These results show that log-concavity need not hold for IDP polytopes, informing the search for precise conditions under which -vectors are unimodal or log-concave; additional details and more examples will be provided in future work.

Abstract

Lattice polytopes are called IDP polytopes if they have the integer decomposition property, i.e., any lattice point in a th dilation is a sum of lattice points in the polytope. It is a long-standing conjecture whether the numerator of the Ehrhart series of an IDP polytope, called the -polynomial, has a unimodal coefficient vector. In this preliminary report on research in progress we present examples showing that -vectors of IDP polytopes do not have to be log-concave. This answers a question of Luis Ferroni and Akihiro Higashitani. As this is an ongoing project, this paper will be updated with more details and examples in the near future.

Paper Structure

This paper contains 1 section, 3 theorems, 6 equations, 1 figure.

Table of Contents

  1. Counterexamples to the log-concavity question

Key Result

Theorem 1.2

Let $P \subset \mathbb{R}^7$ be the lattice polytope given by the convex hull of $e_1, \ldots, e_7$ and the following two vectors and Then $P$ has the following properties:

Figures (1)

  • Figure 1: A bipartite graph realising the $12$-dimensional example.

Theorems & Definitions (4)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4