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!



Zeros of linear partial fractions
$${1\over z+1}+{-2\over z+10}+{1\over z+20}={170-z\over(z+1)(z+10)(z+20)}$$ seems to satisfy your conditions with $n=3$ and $M=20$ but the zero at 170 is not in the same interval as the poles. 


