Variational analysis of the abscissa mapping for polynomials (Q2706169)

The abscissa mapping NEWLINE\[NEWLINE a:{\mathcal M}_n\longrightarrow R,\quad a(p)=\max \{ \;\Re \xi \;|\;p(\xi)=0\;\} NEWLINE\]NEWLINE defined on the affine variety \({\mathcal M}_n\) of monic polynomials of degree \(n\) plays a central role in the stability theory of matrices and polynomial systems. It is continuous, but not Lipschitz continuous. By using some techniques of modern nonsmooth analysis, the authors completely characterize the subderivative and the subgradients of the abscissa mapping, and establish that the abscissa mapping is everywhere subdifferentially regular. This regularity permits the application of results in a broad context through the use of standard chain rules for nonsmooth functions. The approach is epigraphical, and the epigraph of the abscissa map is everywhere Clarke regular. An application of these results to study the variational behavior of the spectral abscissa map \(M\mapsto a(\det (\lambda I-M))\) is presented.