Equivariant Gr\"obner bases

TL;DR

This paper reviews the theory and applications of equivariant Gr"obner bases, introduces algorithms for their computation, and discusses software implementation and open challenges in polynomial rings with infinite variables.

Contribution

It develops new algorithms for computing equivariant Gr"obner bases and provides a software implementation, advancing computational methods in infinite-variable polynomial rings.

Findings

Algorithms successfully compute equivariant Gr"obner bases

Software implementation demonstrates practical applicability

Identifies open problems and future computational challenges

Abstract

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 Gr\"obner bases, develop several algorithms to compute them, showcase our software implementation, and close with several open problems and computational challenges.