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 ) ≤ 1 for every P ∈ S(n). Define P ∈ S(n) to be locally extremal if d(P ) ≥ d(Q) for all nearby Q ∈ S(n), and note that identifying all locally extremal polynomials would settle the Sendov conjecture. In this paper, we show that the polynomial