Geometry Seminar

D-companion matrices and

geometry of polynomials

Dr. Tuen Wai Ng

The University of Hong Kong

Abstract

In this talk, we shall introduce a new type of companion matrices, D-companion matrices. By using these D-companion matrices, we are able to apply matrix theory directly to the study of geometry of polynomials. In fact, this new approach will allow us to prove quite a number of old and new results in geometry of polynomials. For example, we shall prove a one parameter family version of Schoenberg-type inequality. We then apply the result to study the relative locations of zeros and critical points of a polynomial and prove a result related to Sendov conjecture for polynomials. The same method also allows us to solve a higher order Schoenberg-type conjecture proposed by M.G. de Bruin and A. Sharma. This is a joint work with W.S. Cheung.

All are welcome