I'm new to poles and zeros, so I'm just looking for some clarity to help my research endeavor. If I'm given an integral of the form
$$\iint \frac{P(u,v)}{Q(u,v)}\,dudv$$
and I'm asked to identify the poles of the integrand on the $u$-plane, then I need to find the roots of
$$Q(u,v) = 0$$
My question is, what happens if the roots of $Q(u,v)$ are rational functions? For example, say
$$Q(u,v) = u(v-1)^2 - (av)^2$$
the poles on the $u$-plane are given by
$$u = \left(\frac{av}{v-1}\right)^2$$
which is a rational expression. Does this pole also have "zeros and poles" of its own? Is there something significant to be said when poles are rational?