On the root invariant regions structure for linear systems

D-decomposition technique is targeted to describe the stability domain inparameter space for linear systems, depending on parameters. The technique isvery simple and effective for the case of one or two parameters. However thegeometry of the arising parameter space decomposition to root invariant regionshas not been studied in detail; it is the purpose of the present paper. Weprove that the number of stability intervals for one real parameter is no morethan n/2 (n being the degree of the characteristic polynomial) and providean example, where this number is achieved. For one complex or two realparameters we estimate the number of root invariant regions (equal n^2-2n+3for complex and 2n^2-2n+3 for real case) and demonstrate that this upperbound is tight. The example with n-1 simply connected stability domains in2D parameter plane is analyzed.