A classification of $n$-tuples of commuting shifts of finite multiplicity

Let $\mathbb{V}$ denote an $n$-tuple of shifts of finite multiplicity, and denote by $\mathrm{A}\mathrm{n}\mathrm{n}(\mathbb{V})$ the ideal consisting of polynomials $p$ in $n$ complex variables such that $p(\mathbb{V})=0$. If $\mathbb{W}$ on $\mathfrak{K}$ is another $n$-tuple of shifts of finite multiplicity, and there is a $\mathbb{W}$-invariant subspace ${\mathfrak{K}}^{\mathrm{\prime }}$ of finite codimension in $\mathfrak{K}$ so that $\mathbb{W}\mathrm{\mid }{\mathfrak{K}}^{\mathrm{\prime }}$ is similar to $\mathbb{V}$, then we write $\mathbb{V}\lesssim \mathbb{W}$. If $$\mathbb{W}$\lesssim $\mathbb{V}$$aswell,thenwewrite$$\mathbb{W}$\approx $\mathbb{V}$$. In the case that $\mathrm{A}\mathrm{n}\mathrm{n}(\mathbb{V})$ is a prime ideal we show that the equivalence class of $\mathbb{V}$ is determined by $\mathrm{A}\mathrm{n}\mathrm{n}(\mathbb{V})$ and a positive integer $k$. More generally, the equivalence class of $\mathbb{V}$ is determined by $\mathrm{A}\mathrm{n}\mathrm{n}(\mathbb{V})$ and an $m$-tuple of positive integers, where $m$ is the number of irreducible components of the zero set of $\mathrm{A}\mathrm{n}\mathrm{n}(\mathbb{V})$.