LAUSR

ABSTRACT Let be a polynomial. The Gauss-Lucas theorem states that its critical points, , are contained in the convex hull of its roots. In a recent quantitative version, Totik shows that if almost all roots are contained in a bounded convex domain , then almost all roots of the derivative are in an neighborhood (in a precise sense). We prove a quantitative version: if a polynomial p has n roots in K and roots outside of K, then has at least n−1 roots in . This establishes, up to a logarithm, a conjecture of the first author: we also discuss an open problem whose solution would imply the full conjecture.