Study of $D$-decompositions by the methods of computational real-valued algebraic geometry

New methods to study the $D$-decomposition with the use of the computational realvalued algebraic geometry were proposed. The number of domains of $D$-decomposition for the polynomial parametric families of polynomials and matrices was estimated. This technique which requires construction of the Gröbner bases and cylindrical decomposition sometimes proves to be more precise than the traditional technique. The symbolic calculation system Maple v.14 and, in particular, its package RegularChains are used.