Maximal and inextensible polynomials and the geometry of the spectra of normal operators

Details Maximal and inextensible polynomials and the geometry of the spectra of normal operators Maximal and inextensible polynomials and the geometry of the spectra of normal operators

2003-09-15

arxiv.org/abs/math/0309233v2

Mathematics

Complex Variables

A condensed version of the first half of this paper appeared in Math. Scand., see arXiv:math/0601600. Parts of the second half

Scientific paper

We consider the set S(n,0) of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ we let $|p|_{0}$ denote the distance from the origin to the zero set of $p'$. We determine all 0-maximal polynomials of degree $n$, that is, all polynomials $p\in S(n,0)$ such that $|p|_{0}\ge |q|_{0}$ for any $q\in S(n,0)$. Using a second order variational method we then show that although some of these polynomials are inextensible, they are not necessarily locally maximal for Sendov's conjecture. This invalidates the recently claimed proofs of the conjectures of Sendov and Smale and shows that the method used in these proofs can only lead to (already known) partial results. In the second part of the paper we obtain a characterization of the critical points of a complex polynomial by means of multivariate majorization relations. We also propose an operator theoretical approach to Sendov's conjecture, which we formulate in terms of the spectral variation of a normal operator and its compression to the orthogonal complement of a trace vector. Using a theorem of Gauss-Lucas type for normal operators, we relate the problem of locating the critical points of complex polynomials to the more general problem of describing the relationships between the spectra of normal matrices and the spectra of their principal submatrices.

Affiliated with

Also associated with

No associations

Rating

Maximal and inextensible polynomials and the geometry of the spectra of normal operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Maximal and inextensible polynomials and the geometry of the spectra of normal operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Maximal and inextensible polynomials and the geometry of the spectra of normal operators will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-511708