In the course of an optimization problem, I encountered this expression $$S = \sum_{i\neq j} \frac{z_i z_j}{(z_i-z_j)^2},$$ where $z_1$, ... , $z_n$ are the roots of a polynomial $f(t)$ of degree $n$.
Do you have any idea on how $S$ may be expressed in terms of $f$? It is clear that the denominator is $\operatorname{disc}(f)^2$, but for the numerator...