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On the zeroes of hypergraph independence polynomials
- Part of
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- Journal:
- Combinatorics, Probability and Computing / Volume 33 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 21 September 2023, pp. 65-84
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- Article
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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.
