I am trying to find some general properties of the zeros of

$P(z) = \sum_{i=1}^n \frac{\alpha_i}{z+z_i}$,

with $\sum_{i} \alpha_i = 0$, $z_i \in [-M\; 0], i=1,\ldots,n$ and all $\alpha_i$ and $z_i$ are real, and $M<\infty$.

Actually, what I really hope to find is that the real part of the zeros lie in the same interval as the poles, i.e., in the interval $[-M \; 0]$.

This problem arises in a particular type of dynamic systems over graphs. The function $P(z)$ is the transfer-function of a linear system running a consensus protocol, so the poles of the system are the eigenvalues of the combinatorial Laplacian.

Thanks!

`$\sum_{i=1}^n\frac{\alpha_i}{z-z_i}$`

? – Emil Jeřábek Feb 25 '11 at 11:23