Borcea’s Variance Conjectures on the Critical Points of Polynomials

Abstract

Closely following recent ideas of J. Borcea, we discuss various modifications and relaxations of Sendov's conjecture about the location of critical points of a polynomial with complex coefficients. The resulting open problems are formulated in terms of matrix theory, mathematical statistics or potential theory. Quite a few links between classical works in the geometry of polynomials and recent advances in the location of spectra of small rank perturbations of structured matrices are established. A couple of simple examples provide natural and sometimes sharp bounds for the proposed conjectures.