Non UBC
DSpace
Eli Shamovich
2019-11-27T09:42:13Z
2019-05-30T17:01
Recall that given two complex polynomials $f$ and $g$, the Bezout matrix $B(f,g) = (b_{ij})$ of $f$ and $g$ is defined by $\frac{f(t)g(s) - f(s)g(t)}{t-s} = \sum_{i,j} b_{ij} t^i s^j$. It is a classical result of Hermite that given a polynomial $p(z)$, the Bezout matrix of $p(z)$ and $p^{\tau}(z) = \overline{p(\bar{z})}$ is skew self-adjoint. The number of common zeroes of $p$ and $p^{\tau}$ is the dimension of the kernel of $-i B(p,p^{\tau})$. Additionally, $p$ has $n_+$ roots in the upper half-plane and $n_-$ in the lower half-plane. Here $n-+$ and $n_-$ stand for the number of positive and negative eigenvalues of $-i B(p,p^{\tau})$, respectively. In this talk, I will describe how one can extend the notion of the Bezout matrix to a pair of meromorphic functions on a compact Riemann surface. If the surface is real and dividing the matrix $-i B(f,f^{\tau})$ is $J$-selfadjoint, for a certain signature matrix $J$. We the study the signature of the Bezoutian and obtain an extension of HermiteÃ¢s theorem to quadrature domains.
This talk is based on joint work with V. Vinnikov.
https://circle.library.ubc.ca/rest/handle/2429/72430?expand=metadata
32.0 minutes
video/mp4
Author affiliation: University of Waterloo
Banff (Alta.)
10.14288/1.0385990
eng
Unreviewed
Vancouver : University of British Columbia Library
Banff International Research Station for Mathematical Innovation and Discovery
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
Postdoctoral
BIRS Workshop Lecture Videos (Banff, Alta)
Mathematics
Operations Research, Mathematical Programming, Algebraic Geometry, Control/Optimization/Operation Research
Counting the number of zeroes of polynomials in quadrature domains
Moving Image
http://hdl.handle.net/2429/72430