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TitleEquivariant Gr\"obner bases and the Gaussian two-factor model.
Author(s) Brouwer Andries E., Jan Draisma
TypeArticle in Journal
AbstractExploiting symmetry in Gröbner 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 Gröbner 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 Gröbner bases.

Using this algorithm and the monoid of strictly increasing functions $ \mathbb{N} \to \mathbb{N}$ we prove that the kernel of the ring homomorphism

$\displaystyle \mathbb{R}[y_{ij} \mid i,j \in \mathbb{N}, i > j] \to\mathbb{R}[s_i,t_i \mid i \in \mathbb{N}], y_{ij} \mapsto s_is_j + t_it_j $

is generated by two types of polynomials: off-diagonal $ 3 \times 3$-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model from algebraic statistics. - See more at: http://www.ams.org/journals/mcom/2011-80-274/S0025-5718-2010-02415-9/home.html#sthash.Lwuz1DXE.dpuf
ISSN0025-5718; 1088-6842/e
URL http://www.ams.org/journals/mcom/2011-80-274/S0025-5718-2010-02415-9/home.html
JournalMath. Comput.
PublisherAmerican Mathematical Society (AMS), Providence, RI
Translation No
Refereed No