Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees
arXiv:0910.1610 · doi:10.1007/s10801-012-0381-1
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.
Section 6 outlines few open problems. (Final version, differs slightly then publication.) Version3 was a major revision: proved Conjecture from previous version for all simplicial complexes