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

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