Maximal and Inextensible Polynomials and the Geometry of the Spectra of Normal Operators

We consider the set S(n, 0) of monic complex polynomials of degree n ≥ 2 having all their zeros in the closed unit disk and vanishing at 0. For p ∈ 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 ∈ S(n, 0) such that |p|0 ≥ |q|0 for any q ∈ 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. Introduction Let Sn be the set of all monic complex polynomials of degree n ≥ 2 having all their zeros in the closed unit disk D̄. If p ∈ Sn and a ∈ Z(p) then the Gauss-Lucas theorem implies that (a+ 2D̄) ∩ Z(p′) 6= ∅, where Z(p) and Z(p′) denote the zero sets of p and p′, respectively. In 1958 Sendov conjectured that this result may be substantially improved in the following way: Conjecture 1. If p ∈ Sn and a ∈ Z(p) then (a+ D̄) ∩ Z(p′) 6= ∅. Sendov’s conjecture is widely regarded as one of the main challenges in the analytic theory of polynomials. Numerous attempts to verify this conjecture have led to over 80 papers, but have met with limited success. We refer to [13], [20] and [21] for surveys of the results on Sendov’s conjecture and related questions. The set Pn of monic complex polynomials of degree n may be viewed as a metric space by identifying it with the quotient ofC by the action of the symmetric group on n elements Σn. Indeed, let τ : C n → C/Σn denote the orbit map. Let further p(z) = ∏n i=1(z − zi) and q(z) = ∏n i=1(z − ζi) be arbitrary polynomials in Pn and set ∆(p, q) = min σ∈Σn max 1≤i≤n |zi − ζσ(i)|. Then ∆ is a distance function on Pn which induces a structure of compact metric space on the set Sn = {p ∈ Pn : ∆(p, z) ≤ 1} = τ(D̄). Conjecture 1 is therefore 2000 Mathematics Subject Classification. Primary: 30C15; Secondary: 47B15.