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Gröbner bases and polyhedral geometry of reducible and cyclic models. (English) Zbl 1044.62065
Summary: This article studies the polyhedral structure and combinatorics of polytopes that arise from hierarchical models in statistics, and shows how to construct Gröbner bases of toric ideals associated to a subset of such models. We study the polytopes for cyclic models, and we give a complete polyhedral description of these polytopes in the binary cyclic case. Further, we show how to build Gröbner bases of a reducible model from the Gröbner bases of its pieces.
This result also gives a different proof that decomposable models have quadratic Gröbner bases. Finally, we present the solution of a problem posed by M. Vlach [Discrete Appl. Math. 13, 61–78 (1986; Zbl 0601.90105)], concerning the dimension of fibers coming from models corresponding to the boundary of a simplex.

62H17 Contingency tables
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
Full Text: DOI
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