The topology of D-stability

The paper is devoted to the study of topological properties of different definitions of stability for polynomials. A new geometric approach based on unification of Dstability approach, canonical parameter space approach and D-decomposition approach to the unconstrained parametric stability problem is presented. Topology and adjacencies the sets of monic complex polynomials with fixed distributions of roots relative to an arbitrary semialgebraic stability region Ω are described. These results are obtained through the application of topological theory of symmetric products based on analysis of correspondence between roots and coefficients of an indeterminate polynomial.