  In this paper we investigate the distribution of zeros of the independence
polynomial of hypergraphs of maximum degree $\Delta$. For graphs the largest
zero-free disk around zero was described by Shearer as having radius
$\lambda_s(\Delta)=(\Delta-1)^{\Delta-1}/\Delta^\Delta$. Recently it was shown
by Galvin et al. that for hypergraphs the disk of radius $\lambda_s(\Delta+1)$
is zero-free; however, it was conjectured that the actual truth should be
$\lambda_s(\Delta)$. We show that this is indeed the case. We also show that
there exists an open region around the interval
$[0,(\Delta-1)^{\Delta-1}/(\Delta-2)^\Delta)$ that is zero-free for hypergraphs
of maximum degree $\Delta$, 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 $\Delta$.
