Continuity argument revisited: geometry of root clustering via symmetric products

Abstract

We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region $\Omega$, on its border, and at the complement to its closure. Presented approach is a generalisation, unification and development of several classical approaches to stability problems in control theory: root clustering ($D$-stability) developed by R.E. Kalman, B.R. Barmish, S. Gutman et al., $D$-decomposition(Yu.I. Neimark, B.T. Polyak, E.N. Gryazina) and universal parameter space method(A. Fam, J. Meditch, J.Ackermann). Our approach is based on the interpretation of correspondence between roots and coefficients of a polynomial as a symmetric product morphism. We describe the topology of strata up to homotopy equivalence and, for many important cases, up to homeomorphism. Adjacencies between strata are also described. Moreover, we provide an explanation for the special position of classical stability problems: Hurwitz stability, Schur stability, hyperbolicity.

45 pages, 4 figures

References in corpus (10)

• Linear Matrix Inequality Representation of Sets
• Classifying real polynomial pencils
• Some remarks about equations defining coincident root loci
• Plane geometry and convexity of polynomial stability regions
• How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties
• Geometry of the symmetrized polydisc
• Fundamental Group and Euler Characteristic of Permutation Products and Fat Diagonals
• Symmetric Squares of Graphs
• The homology of spaces of polynomials with roots of bounded multiplicity
• Hölder estimates for Cauchy-Type Integrals and proper holomorphic mappings of symmetric products