Aims and Scope:
Academician Blagovest Sendov raised in 1958 the following
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.
University of Cyprus, New Campus, Nicosia, Cyprus