Variational analysis of the abscissa mapping for polynomials

The abscissa mapping on the a ne variety M-n of monic polynomials of degree n is the mapping that takes a monic polynomial to the maximum of the real parts of its roots. This mapping plays a central role in the stability theo ry of matrices and dynamical systems. It is well known that the abscissa ma pping is continuous on M-n, but not Lipschitz continuous. Furthermore, its natural extension to the linear space P-n of polynomials of degree n or les s is not continuous. In our analysis of the abscissa mapping, we use techni ques of modern nonsmooth analysis described extensively in Variational Anal ysis (R. T. Rockafellar and R. J.-B. Wets, Springer-Verlag, Berlin, 1998). Using these tools, we completely characterize the subderivative and the sub gradients of the abscissa mapping, and establish that the abscissa mapping is everywhere subdifferentially regular. This regularity permits the applic ation of our results in a broad context through the use of standard chain r ules for nonsmooth functions. Our approach is epigraphical, and our key res ult is that the epigraph of the abscissa map is everywhere Clarke regular.