Universal conditions on $h^{\ast }$-vectors of lattice simplices

Abstract

In this paper, we will prove that given a lattice simplex with its $h^{\ast }$-polynomial ${\sum }_{i\ge 0}{h}_i^{\ast }{t}^i$, if $h_{k+1}^{\ast }=\cdots =h_{2k}^{\ast }=0$ holds, then there exists a lattice simplex of degree $k$ whose $h^{\ast }$-polynomial coincides with ${\sum }_{i=0}^k{h}_i^{\ast }{t}^i$. Moreover, we will present the examples showing that the condition $h_{k+1}^{\ast }=h_{k+2}^{\ast }=\cdots =h_{2k-1}^{\ast }=0$ is necessary.

arXiv comment7 pages