Computability of Equivariant Gr\"obner bases

TL;DR

This paper proves that Gr"obner bases for equivariant ideals are computable and ideal membership is decidable under certain conditions, but also identifies conditions leading to undecidability.

Contribution

It establishes computability of equivariant Gr"obner bases and ideal membership, and provides criteria for undecidability in cases lacking the Hilbert basis property.

Findings

Equivariant Gr"obner bases are computable under Hilbert basis property.

Ideal membership is decidable for equivariant ideals with finitely generated structure.

Conditions for undecidability are identified for common non-Hilbert basis cases.

Abstract

Let $\mathbb{K}$ be a field, $\mathcal{X}$ be an infinite set (of indeterminates), and $\mathcal{G}$ be a group acting on $\mathcal{X}$. An ideal in the polynomial ring $\mathbb{K}[\mathcal{X}]$ is called equivariant if it is invariant under the action of $\mathcal{G}$. We show Gr\"obner bases for equivariant ideals are computable are hence the equivariant ideal membership is decidable when $\mathcal{G}$ and $\mathcal{X}$ satisfies the Hilbert's basis property, that is, when every equivariant ideal in $\mathbb{K}[\mathcal{X}]$ is finitely generated. Moreover, we give a sufficient condition for the undecidability of the equivariant ideal membership problem. This condition is satisfied by the most common examples not satisfying the Hilbert's basis property.