Geodesic
Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees
Journal of Algebraic Combinatorics, Tome 37 (2013) no. 4, pp. 667-682.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Keywords: Stanley-resiner ideal, Betti numbers, Markov basis, Markov width, toric ideal generators 
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     title = {Betti numbers of {Stanley-Reisner} rings determine hierarchical {Markov} degrees},
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Petrović, Sonja; Stokes, Erik. Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees. Journal of Algebraic Combinatorics, Tome 37 (2013) no. 4, pp. 667-682. http://geodesic.mathdoc.fr/item/JAC_2013__37_4_a5/