Equivariant Gröbner bases

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

Theorems of Carathéodory, Minkowski-Weyl, and Gordan up to symmetry

“…One prominent theme in this theory is to find equivariant versions of known results related to ideals in Noetherian polynomial rings. Examples of successful extensions include equivariant Hilbert's basis theorem [2,4,5,9,18], equivariant Hilbert-Serre theorem [11,16,17], equivariant Buchberger algorithm [8], equivariant Hochster's formula [15]. See, e.g., also [7,12,13,14,19,20,21] for related results.…”

Components of symmetric wide-matrix varieties

“…These Noetherianity results admit proofs using Gröbner methods in the spirit of [SS17], which can be turned into explicit algorithms. A special-purpose algorithm was used in [BD11] to find the defining equations for the Gaussian two-factor model, a general-purpose algorithm was implemented in Macaulay 2 [HKL13]. The results of the current paper are also effective: there exists an algorithm that, on input the finitely many equations defining a wide-matrix scheme, computes the quasipolynomial from the Main Theorem.…”

Noetherianity and Specht problem for varieties of bicommutative algebras

“…Again, as in Remarks 3.3 and 4.2 we can ask about the number of generators of the T-ideals of F (B) when the base field is of positive characteristic. Since we work in the polynomial algebra K[Y, Z] considered as a K[X]-bimodule we shall mention several results concerning the number of generators, theory of Gröbner bases and other algorithmic problems: Aschenbrenner, Hillar [2], Hillar, Windfeldt [20], Hillar, Sullivant [19], Krone [24], and Hillar, Krone, Leykin [18].…”