Search for: All records

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$$k\ge 3$$,$$k$$-uniformlinearhypergraphs have a much larger zero-free disc of radius$$\Omega (\Delta ^{- \frac{1}{k-1}} )$$. We establish this in the case of linear hypertrees.