Let $f(z)=\sum\limits_{k=0}^N a_kz^k$ be a (monic) complex polynomial and $\{\xi_{k}\}_{k=1}^{N}$ be the roots of $f$ (with multiplicities). Let $\{\tilde{\xi_{k}}\}_{k=1}^{N}$ be the perturbed version of $\{\xi_{k}\}_{k=1}^{N}$ by a "small" perturbation and $\tilde{f}(z)=\sum\limits_{k=0}^N \tilde{a}_k z^k$ be the (monic) complex polynomial with the roots $\{\tilde{\xi}_{k}\}_{k=1}^{N}$.
How do the difference between $\{a_k \}_{k=1}^{N}$ and $\{\tilde{a}_k \}_{k=1}^{N}$ relate quantitatively to the perturbation of $\{\xi_{k}\}_{k=1}^{N}$ (possibly in $l^2$ or $l^{\infty}$) ? I know that $\{a_k \}_{k=1}^{N}$ and $\{\tilde{a}_k \}_{k=1}^{N}$ should be qualitatively "near" to each other, since as is known the viete map is continuous.
Is there any paper/book which deals with this problem?
I would appreciate for any help.
$a_{N-k} = \sum_S \prod_{j \in S} (-\xi_j)$, the sum being over all subsets of $\{1,\ldots,N\}$ of cardinality $k$. So $$|a_{N-k} - \tilde{a}_{N-k}| \le \sum_S \left|\prod_{j \in S} \xi_j - \prod_{j \in S} \tilde{\xi}_j\right| $$ Now if all $|\xi_j| \le M$ and all $|\tilde{\xi}_j| \le M$, $$\left|\prod_{j \in S} \xi_j - \prod_{j \in S} \tilde{\xi}_j\right| \le M^{k-1} \sum_{j \in S} |\xi_j - \tilde{\xi}_j|$$ Thus $$|a_{N-k} - \tilde{a}_{N-k}| \le k \;\max(\|\xi\|_\infty, \|\tilde{\xi}\|_\infty)^{k-1} \;\|\xi - \tilde{\xi}\|_1$$ So, for example, if $\max(\|\xi\|_\infty, \|\tilde{\xi}\|_\infty) < 1$ we have $$\|a - \tilde{a}\|_1 < \dfrac{\|\xi - \tilde{\xi}\|_1}{\left(1 - \max(\|\xi\|_\infty, \|\tilde{\xi}\|_\infty)\right)^2}$$ and $$ \|a - \tilde{a}\|_\infty \le - \dfrac{\|\xi - \tilde{\xi}\|_1}{e \max(\|\xi\|_\infty, \|\tilde{\xi}\|_\infty) \ln \max(\|\xi\|_\infty, \|\tilde{\xi}\|_\infty)}$$