Advanced Studies in Pure Mathematics

Equivariant Gröbner bases

Christopher J. Hillar, Robert Krone, and Anton Leykin

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Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible the development of effective routines. Ability to compute relies on finite generation up to symmetry for ideals invariant under a large group or monoid action, such as the permutations of the natural numbers. We summarize the current state of theory and applications for equivariant Gröbner bases, develop several algorithms to compute them, showcase our software implementation, and close with several open problems and computational challenges.

Article information

The 50th Anniversary of Gröbner Bases, T. Hibi, ed. (Tokyo: Mathematical Society of Japan, 2018), 129-154

Received: 5 October 2016
Revised: 31 March 2017
First available in Project Euclid: 21 September 2018

Permanent link to this document euclid.aspm/1537499601

Digital Object Identifier

Primary: 13E05: Noetherian rings and modules 13E15: Rings and modules of finite generation or presentation; number of generators 20B30: Symmetric groups 06A07: Combinatorics of partially ordered sets

Invariant ideal well-quasi-ordering symmetric group Gröbner basis generating sets infinite dimensional algebra


Hillar, Christopher J.; Krone, Robert; Leykin, Anton. Equivariant Gröbner bases. The 50th Anniversary of Gröbner Bases, 129--154, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07710129.

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