Stability Of Matrix Polynomials In One And Several Variables

The paper presents methods of eigenvalue localisation of regular matrix polynomials, in particular, stability of matrix polynomials is investigated. For this aim a stronger notion of hyperstability is introduced and widely discussed. Matrix versions of the Gauss-Lucas theorem and Sz\'asz inequality are shown. Further, tools for investigating (hyper)stability by multivariate complex analysis methods are provided. Several second- and third-order matrix polynomials with particular semi-definiteness assumptions on coefficients are shown to be stable.