Is it impossible to determine the inverse Z-transform without any other information? Suppose I give you this as my transfer function $H(z)$:

$$   H(z) = \frac{1} { 1 - az^{-1}}$$

With no other information given, is it even possible to determine the inverse Z-transform?

The reason I'm asking is because on a particular Z-transform table I have, there are two possible resolutions for this, depending on the desired region of convergence:

 

As one can see we have two choices,


$a^nu[n]$
$-a^nu[-n-1]$


So I think I need more information about the system to make the right decision, that is, whether the response is supposed to be causal, stable, or both.
It's true that the algebraic expression for the $\mathcal{Z}$-transform is generally not sufficient for computing the corresponding time-domain sequence. The additional information we need is the region of convergence (ROC). The ROC together with the expression for the $\mathcal{Z}$-transform uniquely determines the corresponding time-domain sequence.

For the given example

$$H(z)=\frac{1}{1-az^{-1}}\tag{1}$$

it is clear that the expression $(1)$ can be obtained in two ways. First,

$$\begin{align}-\sum_{n=-\infty}^{-1}a^nz^{-n}&amp;=-\sum_{n=1}^{\infty}a^{-n}z^n\\&amp;=-\frac{\frac{z}{a}}{1-\frac{z}{a}}\\&amp;=\frac{1}{1-az^{-1}},\quad |z|&lt;|a|\end{align}$$

corresponding to the sequence $h[n]=-a^nu[-n-1]$, and, secondly,

$$\sum_{n=0}^{\infty}a^nz^{-n}=\frac{1}{1-az^{-1}},\quad |z|&gt;|a|$$

corresponding to $h[n]=a^nu[n]$. In both cases I've used the formula for the geometric series.

The only difference is the ROC, so it's crucial to know the ROC in addition to the expression for the $\mathcal{Z}$-transform. The ROC $|z|&gt;|a|$ indicates a right-sided sequence, whereas the ROC $|z|&lt;|a|$ corresponds to a left-sided sequence. For higher order systems, you can also have an annulus $|a|&lt;|z|&lt;|b|$ as ROC, corresponding to a two-sided sequence.

If the sequence is interpreted as the impulse response of a system, then the system is stable if the ROC contains the unit circle $|z|=1$. It is causal if the sequence is zero for $n&lt;0$.