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.Comment: 10 pages, AMS-LaTeX, 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
