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In arXiv:1709.07504 Ardila and Aguiar give a Hopf monoid structure onhypergraphs as well as a general construction of polynomial invariants on Hopfmonoids. Using these results, we define in this paper a new polynomialinvariant on hypergraphs. We give a combinatorial interpretation of thisinvariant on negative integers which leads to a reciprocity theorem onhypergraphs. Finally, we use this invariant to recover well-known invariants onother combinatorial objects (graphs, simplicial complexes, building sets etc)as well as the associated reciprocity theorems.

The Hopf monoid of hypergraphs and its sub-monoids: basic invariant and reciprocity theorem