Title: Perturbations of roots of polynomials

Abstract: Let ${\mathcal P}_n$ be the vector space of all polynomials of degree $\leq n$. Let $A$ be a linear transformation on ${\mathcal P}_n$. The following question will be discussed:

Can we identify a region in the complex plane which contains all the roots of the polynomial $Ap$, provided that all the roots of a polynomial $p \in {\mathcal P}_n$ are given?

The celebrated Gauss-Lucas theorem answers this question when $A(p) = p'$. A theorem of Cauchy addresses a special case when $A$ is a linear combination of derivatives and $p(x) = x^n$. Improvements of each of these results, as well as new results identifying a class of transformations $A$ for which the above question has a nice answer will be presented.