Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees

摘要

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.