Optimal zero-free regions for the independence polynomial of bounded degree hypergraphs

In this paper we investigate the distribution of zeros of the independence polynomial of hypergraphs of maximum degree $\mathrm{\Delta }$. For graphs the largest zero-free disk around zero was described by Shearer as having radius ${\lambda }_s(\mathrm{\Delta })=(\mathrm{\Delta }-1{)}^{\mathrm{\Delta }-1}\mathrm{/}{\mathrm{\Delta }}^{\mathrm{\Delta }}$. Recently it was shown by Galvin et al. that for hypergraphs the disk of radius ${\lambda }_s(\mathrm{\Delta }+1)$ is zero-free; however, it was conjectured that the actual truth should be ${\lambda }_s(\mathrm{\Delta })$. We show that this is indeed the case. We also show that there exists an open region around the interval $[0,(\mathrm{\Delta }-1{)}^{\mathrm{\Delta }-1}\mathrm{/}(\mathrm{\Delta }-2{)}^{\mathrm{\Delta }})$ that is zero-free for hypergraphs of maximum degree $\mathrm{\Delta }$, which extends the result of Peters and Regts from graphs to hypergraphs. Finally, we determine the radius of the largest zero-free disk for the family of bounded degree $k$-uniform linear hypertrees in terms of $k$ and $\mathrm{\Delta }$.