ON THE LOCATION OF CRITICAL POINTS OF SOME COMPLEX POLYNOMIALS

TL;DRAbstract

Let P(a, n) be the set of all complex polynomials of degree n which have all their roots in the closed unit disk and one fixed root at a, 0 < a < 1. In this paper we show that for n > 3 all critical points of the polynomial f(z) = (z n_1 + l)(z -a) lie outside the set K{a, n) consisting of all b such that for some c the polynomial p(z) = (z -b) n -c belongs to V(a,n). Hence we infer that minimal sets satisfying the Sendov property (i.e. containing at least one critical point of each p 6 V(a, n)) exist but they are not unique.

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