Borcea's variance conjectures on the critical points of polynomials

Details Borcea's variance conjectures on the critical points of polynomials Borcea's variance conjectures on the critical points of polynomials

2010-10-25

arxiv.org/abs/1010.5167v2

Mathematics

Complex Variables

Scientific paper

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.

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