Continuity argument revisited: geometry of root clustering via symmetric products

TL;DR

This paper explores the topology of polynomial root clustering spaces using symmetric products, unifying classical stability analysis methods in control theory and describing the structure and adjacency of stability regions.

Contribution

It introduces a unified geometric framework based on symmetric products to analyze stability regions and their topological properties in polynomial root spaces.

Findings

Topology of polynomial stratification described up to homotopy and homeomorphism

Adjacency relations between stability strata characterized

Unified geometric approach to classical stability problems

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.