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arXiv · 0910.1610

Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees

Sonja Petrović·Erik Stokes
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Abstract

There are two seemingly unrelated ideals associated with a simplicial complex Δ. One is the Stanley-Reisner ideal I_Δ, the monomial ideal generated by minimal non-faces of Δ, well-known in combinatorial commutative algebra. The other is the toric ideal I_{M(Δ)} of the facet subring of Δ, whose generators give a Markov basis for the hierarchical model defined by Δ, playing a prominent role in algebraic statistics. In this note we show that the complexity of the generators of I_{M(Δ)} is determined by the Betti numbers of I_Δ. The unexpected connection between the syzygies of the Stanley-Reisner ideal and degrees of minimal generators of the toric ideal provide a framework for further exploration of the connection between the model and its many relatives in algebra and combinatorics.

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Sonja Petrović, Erik Stokes. 2012-05-31. Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees. https://doi.org/10.1007/s10801-012-0381-1

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