We study the locations of complex zeroes of independence polynomials of
bounded degree hypergraphs. For graphs, this is a long-studied subject with
applications to statistical physics, algorithms, and combinatorics. Results on
zero-free regions for bounded-degree graphs include Shearer’s result on the
optimal zero-free disk, along with several recent results on other zero-free
regions. Much less is known for hypergraphs. We make some steps towards an
understanding of zero-free regions for bounded-degree hypergaphs by proving
that all hypergraphs of maximum degree $\Delta$ have a zero-free disk almost as
large as the optimal disk for graphs of maximum degree $\Delta$ established by
Shearer (of radius $\sim 1/(e \Delta)$). Up to logarithmic factors in $\Delta$
this is optimal, even for hypergraphs with all edge-sizes strictly greater than
$2$. We conjecture that for $k\ge 3$, $k$-uniform linear hypergraphs have a
much larger zero-free disk of radius $\Omega(\Delta^{- \frac{1}{k-1}} )$. We
establish this in the case of linear hypertrees.
