ROC of Z transform Doesnt include a pole on the boundary?
The ROC $0.5&lt;|z|&lt;\infty$ does not imply that there's a pole at $z=0.5$. What it does say is that there's at least one pole satisfying $|z|=0.5$ (and no other pole with a radius larger than $0.5$), and this is also the case for the given denominator polynomial. The roots of the polynomial are
$$z_{1,2}=0.5\, e^{j\pi/3}\tag{1}$$
and
$$z_{3,4}=0.5\, e^{-j\pi/3}\tag{1}$$
I.e., there are two double poles, and all four poles lie on a circle with radius $0.5$.