The Gauss-Lucas theorem in an asymptotic sense (Q2830657)

The Gauss-Lucas theorem states that the convex hull of the zeros of a univariate polynomial contains the zeros of its derivative. In particular, if all of the zeros of a polynomial lie within a convex set \(K\), then all of the zeros of its derivative also lie in \(K\). The paper describes one possible extension of this result to sequences of polynomials of increasing degree. The result states that if the polynomials in a sequence have almost all of their zeros in a compact convex set \(K\), then for every \(\varepsilon>0\), their derivative polynomials have almost all of their zeros in an \(\varepsilon\) neighborhood of \(K\). In this statement, ``almost all of the zeros in a set'' means that the ratio of the number of zeros inside the set and the number of zeros outside of the set tends to zero when the degree tends to infinity.