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Mathematics of Computation

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Equivariant Gröbner bases and the Gaussian two-factor model


Authors: Andries E. Brouwer and Jan Draisma
Journal: Math. Comp. 80 (2011), 1123-1133
MSC (2010): Primary 13P10, 16W22; Secondary 62H25
DOI: https://doi.org/10.1090/S0025-5718-2010-02415-9
Published electronically: September 9, 2010
MathSciNet review: 2772115
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Abstract: Exploiting 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.


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Additional Information

Andries E. Brouwer
Affiliation: Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Email: aeb@cwi.nl

Jan Draisma
Affiliation: Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands and Centrum voor Wiskunde en Informatica, Amsterdam, The Netherlands
Email: j.draisma@tue.nl

DOI: https://doi.org/10.1090/S0025-5718-2010-02415-9
Keywords: Equivariant Gröbner bases, algebraic factor analysis
Received by editor(s): August 11, 2009
Received by editor(s) in revised form: February 3, 2010
Published electronically: September 9, 2010
Additional Notes: The second author is supported by DIAMANT, an NWO mathematics cluster.
Article copyright: © Copyright 2010 A. E. Brouwer and J. Draisma

American Mathematical Society