Do a pole/zero plot and specified ROC uniquely define an inverse z-transform?

Question

I know that if I have a closed-form algebraic expression $X(z)$ and I specify the region of convergence, this uniquely identifies exactly time-domain sequence (inverse Z-transform) $x[n]$.

Let's suppose I have a pole-zero plot specifying that $X(z)$ has a single zero at $z=-1$ and four poles at $z=1/2, 2, 3,$ and $4$ (just for the sake of a concrete example). However, I have not specified an algebraic form of $X(z)$. Let's suppose that I then specify the region of convergence as $2<|z|<3$. Am I justified in saying that there is exactly one time-domain sequence $x[n]$ that corresponds to this pole-zero plot and ROC?

Answer 1

HINT:

Note that the poles and zeros of

$$X(z)=c\cdot\frac{N(z)}{D(z)}$$

only depend on the polynomials $N(z)$ and $D(z)$. Consequently, pole and zero locations determine the transfer function only up to a scalar constant.