On the numerical condition of a generalized Hankel eigenvalue problem

The generalized eigenvalue problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde H y \,{=}\, \lambda H y$$\end{document} with H a Hankel matrix and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde H$$\end{document} the corresponding shifted Hankel matrix occurs in number of applications such as the reconstruction of the shape of a polygon from its moments, the determination of abscissa of quadrature formulas, of poles of Padé approximants, or of the unknown powers of a sparse black box polynomial in computer algebra. In many of these applications, the entries of the Hankel matrix are only known up to a certain precision. We study the sensitivity of the nonlinear application mapping the vector of Hankel entries to its generalized eigenvalues. A basic tool in this study is a result on the condition number of Vandermonde matrices with not necessarily real abscissas which are possibly row-scaled.