Determine impulse response given input and output: which ROC?
The function $H(z)$ without a specified ROC is generally not sufficient for uniquely determining the time domain sequence $h[n]$. You need more information about the desired $h[n]$. There are three types of ROCs:


the ROC is an annulus ($R_1&lt;|z|&lt;R_2$): $h[n]$ is a two-sided sequence (non-causal)
the ROC is defined by $|z|&gt;R$: $h[n]$ is a right-sided sequence
the ROC is defined by $|z|&lt;R$: $h[n]$ is a left-sided sequence


If you're looking for a stable system, then the ROC needs to include the unit circle $|z|=1$. If that does not uniquely define the ROC, then you need to know if you're looking for a left-sided, a right-sided, or a two-sided sequence.