On the Hausdorff Geometry of Polynomials
by
Bl. Sendov
Bulgarian Academy of Science, Sofia

A set of n points A = {z₁, z₂, ..., z_n} on the complex plane define the monic polynomial p(z) = (z-z₁)(z-z₂) ... (z-z_n) with derivative p'(z) = n(z-\zeta₁)(z-\zeta₂) ... (z-\zeta_n-1). The Geometry of polynomials studies the relations between the sets A and A' = {\zeta₁, \zeta₂, ..., \zeta_n-1}. The classical Gauss-Lucas Theorem asserts that the convex hull of A contains A'.

In 1958 we formulated the following conjecture in the Geometry of Polynomials.

Conjecture. If all the zeros of the polynomial p(z) = (z-z₁)(z-z₂) ... (z-z_n)  (n >= 2) lie in the unit disk D = D(0, 1) = {z:  |z| <= 1}, then for every z_k the disk D(z_k, 1) = {z:  |z-z_k| <= 1} contains at least one zero of p'(z).

Up to now, Conjecture 1 was neither proved nor disproved, although more than 80 related papers have been published on this topic. In attempts to prove this conjecture, many new results have been obtained and many other problems and conjectures have been formulated. Our purpose in this lecture is to review some of these results and to formulate some new problems from the point of view of Hausdorff distance between the set of zeros and the set of critical points of a polynomial.

http://copern.bas.bg/~bsendov/

Date received: August 2, 2001