Computability of Equivariant Gr\"obner bases

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.