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

## TL;DR

This paper establishes universal conditions on the $h^*$-vectors of lattice simplices, proving existence results under certain vanishing conditions and providing examples to show the necessity of these conditions.

## Contribution

It introduces new conditions on $h^*$-vectors of lattice simplices and proves existence results, advancing understanding of their structure.

## Key findings

- Existence of lattice simplices with prescribed $h^*$-vector segments.
- Necessary conditions for vanishing $h^*$-vector entries.
- Examples demonstrating the sharpness of the conditions.

## Abstract

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.