PIMS / AMI Seminar: Blagovest Sendov

Let D(c; r) be the smallest disk, with center c and radius r, containing all zeros of the polynomial p(z) = (z–z1)(z–z2) · · · (z–zn). In 1958, we conjectured that for every zero zk of p(z), the disk D(zk; r) contains at least one zero of the derivative p¢(z). More than 100 papers are devoted to this conjecture, proving it for different special cases. But in general, the conjecture is proved only for the polynomials of degree n≤8. In this lecture we review the latest developments and generalizations of the conjecture.