I was wondering if there is anything known about the geometry (position) of the roots of a rational map of the form $$R(z):= \sum_{j=1}^{n} \frac{a_j}{z-p_j},$$ where the $a_j$'s are nonzero complex numbers and the $p_j$'s are distinct complex numbers.

Any relevant reference is welcome.

Note : The Gauss-Lucas theorem implies that if the $a_j$'s are all assumed to be positive, then the roots of $R$ lie in the convex hull of the set of poles $\{p_1,\dots, p_n\}$. However, this is false in the general case, but perhaps there is something to be said.

Thank you, Malik