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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Equivariant Gröbner bases and the Gaussian two-factor model
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by Andries E. Brouwer and Jan Draisma PDF
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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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
  • 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