BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Counting the number of zeroes of polynomials in quadrature domains Shamovich, Eli


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 Hermiteâs theorem to quadrature domains. This talk is based on joint work with V. Vinnikov.

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