The Abscissa Mapping for Polynomials and the Gauss-Lukas Theorem James Burke University of Washington Adrian Lewis Simon Fraser University Michael Overton New York University

AMS Classification: 34D05, 34D20, 49J52 Keywords: nonsmooth analysis, polynomials, stability, subgradient

Abstract: Recently Burke and Overton have shown that the mapping of a polynomial to the maximum real part of its roots (the abscissa mapping) is everywhere subdifferentially regular even though this mapping itself is non-Lipschitzian. Their analysis relies on earlier work due to Levantovskii. Unfortunately, the proof is carried forth by brute force. It is extremely lengthly, arduous, and opaque, and does not generalize to other mappings of the roots. In this paper we present a more elementary proof based on the Gauss-Lukas Theorem. This proof better illustrates the geometric essentials and thereby holds the promise of being the key to understanding the variational geometry of similar mappings of polynomials. References:

1. “Variational analysis of the abscissa mapping for polynomials”, James Burke and Michael Overton, SIAM J. Control and optimization, 39(2001), 1651–1676.

2. “Variational analysis of non-Lipschitz spectral functions”, James Burke and Michael Overton, Mathematical Programming, 90(2001), 317–351.