On Properties of Semipositive Cones and Simplicial Cones

For a given nonsingular $n\times n$ matrix $A$, the cone $S_{A}=\{x:Ax\geq 0\}$ , and its subcone $K_A$ lying on the positive orthant, called as semipositive cone, are considered. If the interior of the semipositive cone $K_A$ is not empty, then $A$ is named as semipositive matrix. It is known that $K_A$ is a proper polyhedral cone. In this paper, it is proved that $S_{A}$ is a simplicial cone and properties of its extremals are analyzed. An one-one relation between simplicial cones and invertible matrices is established. For a proper cone $K$ in $\mathbb{R}^n$, $\pi(K)$ denotes the collection of $n\times n$ matrices that leave $K$ invariant. For a given minimally semipositive matrix (no column-deleted submatrix is semipositive) $A$, it is shown that the invariant cone $\pi(K_A)$ is a simplicial cone.

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.