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Associated to any hypergraph is a toric ideal encoding the algebraicrelations among its edges. We study these ideals and the combinatorics of theirminimal generators, and derive general degree bounds for both uniform andnon-uniform hypergraphs in terms of balanced hypergraph bicolorings,separators, and splitting sets. In turn, this provides complexity bounds foralgebraic statistical models associated to hypergraphs. As two mainapplications, we recover a well-known complexity result for Markov bases ofarbitrary 3-way tables, and we show that the defining ideal of the tangentialvariety is generated by quadratics and cubics in cumulant coordinates.

Combinatorial degree bound for toric ideals of hypergraphs