Variational Analysis of the Abscissa Mapping for Polynomials via the Gauss-Lucas Theorem

Consider the linear space n of polynomials of degree n or less over the complex field. The abscissa mapping on n is the mapping that takes a polynomial to the maximum real part of its roots. This mapping plays a key role in the study of stability properties for linear systems. Burke and Overton have shown that the abscissa mapping is everywhere subdifferentially regular in the sense of Clarke on the manifold n of polynomials of degree n. In addition, they provide a formula for the subdifferential. The result is surprising since the abscissa mapping is not Lipschitzian on . A key supporting lemma uses a proof technique due to Levantovskii for determining the tangent cone to the set of stable polynomials. This proof is arduous and opaque. It is a major obstacle to extending the variational theory to other functions of the roots of polynomials. In this note, we provide an alternative proof based on the Gauss-Lucas Theorem. This new proof is both insightful and elementary.