Unexpected local extrema for the Sendov conjecture
Abstract
Let S(n) be the set of all polynomials of degree n with all roots in the unit disk, and define d(P) to be the maximum of the distances from each of the roots of a polynomial P to that root's nearest critical point. In this notation, Sendov's conjecture asserts that d(P)<=1 for every P in S(n). Define P in S(n) to be locally extremal if d(P)>=d(Q) for all nearby Q in S(n), and note that maximizing d(P) over all locally extremal polynomials P would settle the Sendov conjecture. Prior to now, the only polynomials known to be locally extremal were of the form P(z)=c(z^n+a) with a=1. In this paper, we determine sufficient conditions for real polynomials of a different form to be locally extremal, and we use these conditions to find locally extremal polynomials of this form of degrees 8, 9, 12, 13, 14, 15, 19, 20, and 26.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2005
 DOI:
 10.48550/arXiv.math/0505424
 arXiv:
 arXiv:math/0505424
 Bibcode:
 2005math......5424M
 Keywords:

 Mathematics  Complex Variables;
 30C15
 EPrint:
 10 pages, AMSLaTeX, no figures. The Maple code and results used in this paper are included in the source files. We constructed an unexpected locally extremal polynomial of degree 8 in version 1, then added degrees 12, 14, 20 and 26 in version 2, and degrees 9, 13, 15 and 19 in version 3