Hypergraph independence polynomials with a zero close to the origin

Abstract For each uniformity $k \geq 3$ , we construct $k$ uniform linear hypergraphs $G$ with arbitrarily large maximum degree $\Delta$ whose independence polynomial $Z_G$ has a zero $\lambda$ with $\left \vert \lambda \right \vert = O\left (\frac {\log \Delta }{\Delta }\right )$ . This disproves a recent conjecture of Galvin, McKinley, Perkins, Sarantis, and Tetali.