Hypergraph independence polynomials with a zero close to the origin

Abstract

For each uniformity $k\ge 3$, we construct $k$-uniform linear hypergraphs $G$ with arbitrarily large maximum degree $\hat{I}”$ whose independence polynomial $Z_G$ has a root $\hat{I}»$ with $|\hat{I}»|=O\left(\frac{\log \hat{I}”}{\hat{I}}”\right)$. This disproves a recent conjecture of Galvin, McKinley, Perkins, Sarantis, and Tetali.

5 pages, 1 figures. Correct typos, and incorporate referee feedbacks