Out of curiosity, I'm wondering the solution of a perturbed problem in the form of :
$$p(z)+ap'(z)=0 $$
where $p(z)$ is a complex polynomial of order $N$, $'$ denotes the first derivative.
Additionally, the root of $p(z)$ and its derivative are known.
I wasn't able to find the solution of the general problem, if you have any idea, I'll be glad to hear them,