Borcea’s Variance Conjectures on the Critical Points of Polynomials

Abstract

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. Mathematics Subject Classification (2000). Primary 12D10; Secondary 26C10, 30C10, 15A42, 15B05.

Spectral dissection of finite rank perturbations of normal operators

“…A linear and bounded operator T acting on a Banach space X is called decomposable, if for every finite open cover of its spectrum (28) σ(T ) ⊂ ∪ n j=1 U j , there are T -invariant subspaces X j ⊂ X, 1 ≤ j ≤ n, with the properties (29) X = X 1 + X 2 + . .…”

“…The booming topics of Aleksandrov-Clark measures [38] and the resurrection of Aronszjan-Donoghue theory for matrix valued measures associated to finite rank perturbations of self-adjoint operators [33,21] are two other notable examples. To name only one less known, additional relevant ramification: an apparently non-related open problem of approximation theory, known as Sendov conjecture, can be translated into spectral estimates of rank-two perturbations of normal matrices, see [28].…”

Which Quartic Polynomials Have a Hyperbolic Antiderivative?

“…The relationship between the zeros of a polynomial and those of its derivative has been of significant interest to mathematicians for at least three centuries. Serguei Shimorin has worked in this area [4]. In this paper, we will study polynomials having all of their zeros on the real line; these are sometimes called hyperbolic polynomials.…”