Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees

TL;DR

This paper reveals a surprising link between the Betti numbers of Stanley-Reisner rings and the degrees of generators in hierarchical Markov models, connecting algebraic topology and algebraic statistics.

Contribution

It establishes that the complexity of generators in hierarchical Markov ideals is determined by the Betti numbers of associated Stanley-Reisner ideals, a novel connection in the field.

Findings

Betti numbers determine the degrees of Markov basis generators.

Syzygies of Stanley-Reisner ideals relate to hierarchical model complexity.

Provides a new framework linking algebraic topology and algebraic statistics.

Abstract

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. 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.