Polytopic Vector Research Articles

Ehrhart $f^*$-Coefficients of Polytopal Complexes are Non-negative Integers

The Ehrhart polynomial $L_P$ of an integral polytope $P$ counts the number of integer points in integral dilates of $P$. Ehrhart polynomials of polytopes are often described in terms of their Ehrhart $h^*$-vector (aka Ehrhart $\delta$-vector), which is the vector of coefficients of $L_P$ with respect to a certain binomial basis and which coincides with the $h$-vector of a regular unimodular triangulation of $P$ (if one exists). One important result by Stanley about $h^*$-vectors of polytopes is that their entries are always non-negative. However, recent combinatorial applications of Ehrhart theory give rise to polytopal complexes with $h^*$-vectors that have negative entries.In this article we introduce the Ehrhart $f^*$-vector of polytopes or, more generally, of polytopal complexes $K$. These are again coefficient vectors of $L_K$ with respect to a certain binomial basis of the space of polynomials and they have the property that the $f^*$-vector of a unimodular simplicial complex coincides with its $f$-vector. The main result of this article is a counting interpretation for the $f^*$-coefficients which implies that $f^*$-coefficients of integral polytopal complexes are always non-negative integers. This holds even if the polytopal complex does not have a unimodular triangulation and if its $h^*$-vector does have negative entries. Our main technical tool is a new partition of the set of lattice points in a simplicial cone into discrete cones. Further results include a complete characterization of Ehrhart polynomials of integral partial polytopal complexes and a non-negativity theorem for the $f^*$-vectors of rational polytopal complexes.

Relationships Among Flag f-Vector Inequalities for Polytopes

We examine linear inequalities satisfied by the flag $f$-vectors of polytopes. One source of these inequalities is the toric $g$-vector; convolutions of its entries are non-negative for rational po...

Relationships Among Flag f -Vector Inequalities for Polytopes

We examine linear inequalities satisfied by the flag $f$-vectors of polytopes. One source of these inequalities is the toric $g$-vector; convolutions of its entries are non-negative for rational polytopes. We prove a conjecture of Meisinger about a redundancy in these inequalities. Another source of inequalities is the {\bf cd}-index; among all $d$-polytopes, each {\bf cd}-index coefficient is minimized on the $d$-simplex. We show that not all of the {\bf cd}-index inequalities are implied by the toric $g$-vector inequalities, and that not all of the toric $g$-vector inequalities are implied by the {\bf cd}-index inequalities. Finally, we show that some inequalities from convolutions of {\bf cd}-index coefficients are implied by other {\bf cd}-index inequalities.