# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Equivariant Gröbner bases and the Gaussian two-factor modelHTML articles powered by AMS MathViewer

by Andries E. Brouwer and Jan Draisma
Math. Comp. 80 (2011), 1123-1133

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

References
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• Retrieve articles in Mathematics of Computation with MSC (2010): 13P10, 16W22, 62H25
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• 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
• MR Author ID: 683807
• ORCID: 0000-0001-7248-8250
• Email: j.draisma@tue.nl
• 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.
• © Copyright 2010 A. E. Brouwer and J. 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
• MathSciNet review: 2772115