Participants:
Dmitry Khavinson |
University of South Florida, USA |
Rajesh Pereira |
University of Guelph, Canada |
Edward Saff |
Vanderbilt University, USA |
Blagovest Sendov |
Bulgarian Academy of Sciences, Bulgaria |
Boris Shapiro |
Stockholm University, Sweden |
Nikos Stylianopoulos |
University of Cyprus, Cyprus |
Aims and Scope:
Academician Blagovest Sendov raised in 1958 the following
tantalising question:
Assume that all zeros of a polynomial P with complex
coefficients lie in the closed unit disk.
Is it true that there exists a zero of the derivative in every disk of radius
one centered at a zero of P?
As it stands today, the question remains open, in spite of
concentrated efforts of several groups or individuals. The problem has an
affirmative answer for polynomials of degree at most eight, and for a few
particular geometric configurations (zeros on a line, on a circle, the convex
hull of zeros is a triangle). More frustrating is that all numerical experiments
support an affirmative answer to Sendov conjecture.
It was the late Julius Borcea who freed Sendov conjecture from the sup-norm
estimates and has elaborated during the last decades a more flexible scheme of
attacking the problem by means of probability type entities involving square
summable norms. A few years ago, a group of close collaborators of Borcea
started a systematic study of these new ideas, from converging and complementary
perspectives: potential theory, matrix analysis, analytic theory of polynomials,
probability theory. Very recently Academician Sendov joined them and added to
the puzzle a powerful new concept: the locus of a univariate polynomial.
This workshop is aimed at continuing regular encounters of
that group of researchers. The topics of their investigation is a part of a long
and glorious tradition of elucidating the geometry of critical points of
polynomial maps.
|