Assume $p$ is a degree $n$ polynomial over $\mathbb C$ with roots $x_1,\dots, x_n$. Is there a way to express $$\sum_{j=1}^n f(x_j)$$ for some given function $f$ as a function of the coefficients of $p$?
In case this is not possible in general:
- Can this be done for say $n=3$?
- Does this work if $f$ itself is a polynomial of small degree?
Example: If $f=\text{id}$, then the sum over the roots is equal to a quotient of coefficients of $p$, by Vieta's formula.