Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees

Understand

There are two seemingly unrelated ideals associated with a simplicial complex \Delta.

• One is the Stanley-Reisner ideal I_\Delta, the monomial ideal generated by minimal non-faces of \Delta, well-known in combinatorial commutative algebra.
• The other is the toric ideal I_{M(\Delta)} of the facet subring of \Delta, whose generators give a Markov basis for the hierarchical model defined by \Delta, playing a prominent role in algebraic statistics.
• In this note we show that the complexity of the generators of I_{M(\Delta)} is determined by the Betti numbers of I_\Delta.