Robustly stable multivariate polynomials

We consider stability and robust stability of polynomials with respect to a given arbitrary disjoint decomposition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb C}^{n} = \Gamma \uplus \Lambda }$$\end{document} . A polynomial is called stable if it has no zeros in the region of instability Λ and robustly stable if it is stable and remains so under small variations of its coefficients. Inspired by the article Robust stability of multivariate polynomials. Part 1: Small coefficient perturbations by Kharitonov et al. (Multidimens Syst Signal Process 10(1):21–32, 1999), we generalise some of their results to arbitrary stability decompositions and develop some fundamental results on robustly stable polynomials. The central one of them is a characterisation of robust stability in terms of the stability of several other polynomials, which yields a test for robust stability based on stability tests. Finally, we consider the special situation that the region of instability is a Cartesian product and recover some results for the standard stability decompositions for linear partial differential resp. difference equations with constant coefficients.