On polynomial $n$-tuples of commuting isometries

We extend some of the results of Agler, Knese, and McCarthy [1] to $n$-tuples of commuting isometries for $n>2$. Let $\mathbb{V}=(V_1,\dots ,V_n)$ be an $n$-tuple of a commuting isometries on a Hilbert space and let Ann$(\mathbb{V})$ denote the set of all $n$-variable polynomials $p$ such that $p(\mathbb{V})=0$. When Ann$(\mathbb{V})$ defines an affine algebraic variety of dimension 1 and $\mathbb{V}$ is completely non-unitary, we show that $\mathbb{V}$ decomposes as a direct sum of $n$-tuples $\mathbb{W}=(W_1,\dots ,W_n)$ with the property that, for each $i=1,\dots ,n$, $W_i$ is either a shift or a scalar multiple of the identity. If $\mathbb{V}$ is a cyclic $n$-tuple of commuting shifts, then we show that $\mathbb{V}$ is determined by Ann$(\mathbb{V})$ up to near unitary equivalence, as defined in [1].