Unexpected local extrema for the Sendov conjecture

Details Unexpected local extrema for the Sendov conjecture Unexpected local extrema for the Sendov conjecture

2005-05-19

arxiv.org/abs/math/0505424v3

Mathematics

Complex Variables

10 pages, AMS-LaTeX, no figures. The Maple code and results used in this paper are included in the source files. We constructe

Scientific paper

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.

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