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TitleEquivariant Grobner Bases and The Gaussian Two-Factor Models
Author(s) Andries E. Brouwer, Jan Draisma
TypeArticle in Journal
AbstractExploiting symmetry in Grobner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on
monomials in finitely many variables. This is largely due to the fact that the group elements, being invertible, cannot preserve a term order. By contrast, inspired by work of Aschenbrenner and Hillar, we introduce the concept of equivariant Grobner basis in a setting where a monoid acts by homomorphisms on monomials in potentially infinitely many variables. We require that the action be compatible with a term order, and under some further assumptions derive a Buchberger-type algorithm for computing equivariant Grobner bases.
Using this algorithm and the monoid of strictly increasing functions N → N
we prove that the kernel of the ring homomorphism
R[y_{ij} | i, j ∈ N, i > j] → R[s_{i} , t_{i} | i ∈ N], y_{ij} → s_{i} s_{j} t_{i} t_{j}
is generated by two types of polynomials: off-diagonal 3 × 3-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model from algebraic statistics.
KeywordsEquivariant Gröbner bases, algebraic factor analysis
URL DOI: 10.1090/S0025-5718-2010-02415-9
JournalMathematics of Computation
PublisherAmerican Mathematical Society
NoteElectronically Published on September 9, 2010
Translation No
Refereed Yes