On the zeroes of hypergraph independence polynomials

Abstract

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 disc, 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 disc almost as large as the optimal disc 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 disc of radius $\Omega (\Delta ^{- \frac{1}{k-1}} )$ . We establish this in the case of linear hypertrees.