Equivariant Gröbner bases. (English) Zbl 1411.13029

Hibi, Takayuki (ed.), The 50th anniversary of Gröbner bases. Proceedings of the 8th Mathematical Society of Japan-Seasonal Institute (MSJ-SI 2015), Osaka, Japan, July 1–10, 2015. Tokyo: Mathematical Society of Japan (MSJ). Adv. Stud. Pure Math. 77, 129-154 (2018).
Summary: 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.
For the entire collection see [Zbl 1404.13003].


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