Borcea's variance conjectures on the critical points of polynomials
Complex Variables
2015-03-17 v2
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.
Cite
@article{arxiv.1010.5167,
title = {Borcea's variance conjectures on the critical points of polynomials},
author = {Dmitry Khavinson and Rajesh Pereira and Mihai Putinar and Edward B. Saff and Serguei Shimorin},
journal= {arXiv preprint arXiv:1010.5167},
year = {2015}
}