LAUSR

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