Unexpected local extrema for the Sendov conjecture

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) is at most 1 for every P in S(n). Define P in S(n) to be locally extremal if d(P) is at least d(Q) for all nearby Q in S(n). In this paper, we determine sufficient conditions for real polynomials of degree n with a root strictly between 0 and 1 and a real critical point of order n−3 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.