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 Δ. For graphs, the largest zero-free disk around zero was described by Shearer as having radius λs (Δ) = (Δ−1)^Δ−1/Δ^Δ. Recently, it was shown by Galvin et al. that for hypergraphs the disk of radius λs (Δ + 1) is zero-free; however, it was conjectured that the actual truth should be λs (Δ). We show that this is indeed the case. We also show that there exists an open region around the interval [0, (Δ−1)^Δ−1/ (Δ−2)^Δ) that is zero-free for hypergraphs of maximum degree Δ, 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 Δ.