Universal conditions on $$h^*$$-vectors of lattice simplices

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