Counting and computing regions of $D$-decomposition: algebro-geometric approach

New methods for D-decomposition analysis are presented. They are based on topology of real algebraic varieties and computational real algebraic geometry. The estimate of number of root invariant regions for polynomial parametric families of polynomial and matrices is given. For the case of two parametric family more sharp estimate is proven. Theoretic results are supported by various numerical simulations that show higher precision of presented methods with respect to traditional ones. The presented methods are inherently global and could be applied for studying D-decomposition for the space of parameters as a whole instead of some prescribed regions. For symbolic computations the Maple v.14 software and its package RegularChains are used.