Inequalities for $f^*$-vectors of Lattice Polytopes

TL;DR

This paper investigates inequalities among the coefficients of $f^*$-vectors of lattice polytopes, revealing patterns similar to those in $f$-vector inequalities and identifying conditions for unimodality.

Contribution

It establishes new inequalities for $f^*$-vector coefficients and explores unimodality properties, linking $f^*$-vectors to existing polytope invariants.

Findings

Inequalities similar to $f$-vector inequalities are proven for $f^*$-vectors.

Certain polytope families exhibit unimodality in their $f^*$-vectors.

Existence of polytopes with unimodal $f^*$-vectors sharing the same $h^*$-vector is shown.

Abstract

The Ehrhart polynomial $\text{ehr}_P(n)$ of a lattice polytope $P$ counts the number of integer points in the $n$-th integral dilate of $P$. The $f^*$-vector of $P$, introduced by Felix Breuer in 2012, is the vector of coefficients of $\text{ehr}_P(n)$ with respect to the binomial coefficient basis $ \left\{\binom{n-1}{0},\binom{n-1}{1},...,\binom{n-1}{d}\right\}$, where $d = \dim P$. Similarly to $h/h^*$-vectors, the $f^*$-vector of $P$ coincides with the $f$-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of $f^*$-vectors of polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of $f$-vectors of simplicial polytopes; e.g., the first half of the $f^*$-coefficients increases and the last quarter decreases. Even though $f^*$-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart $h^*$-vector, there is a polytope with the same $h^*$-vector whose $f^*$-vector is unimodal.