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.

Participants:

• Dmitry Khavinson, University of South Florida, USA
• Rajesh Pereira, University of Guelph, Canada
• Boris Shapiro, Stockholm University, Sweden
• Edward Saff, Vanderbilt University, USA
• Blagovest Sendov, Bulgarian Academy of Sciences, Bulgaria
• Nikos Stylianopoulos, University of Cyprus, Cyprus

Workshop website: http://www.math.bas.bg/w-hausdorff/