Hyperbolic matrix polynomials are an important class of Hermitian matrix poly-nomials that contain overdamped quadratics as a special case. They share with definite pencils the spectral property that their eigenvalues are real and semisimple. We extend the definition of hyperbolic matrix polynomial in a way that relaxes the requirement of definiteness of the leading coefficient matrix, yielding...

We show that any symmetric positive definite homogeneous matrix polynomial M ∈ R[x1, . . . , xn] admits a piecewise semi-certificate, i.e. a collection of identites M(x) = P j fi,j(x)Ui,j(x) T Ui,j(x) where Ui,j(x) is a matrix polynomial and fi,j(x) is a non negative polynomial on a semialgebraic subset Si, where R = ∪ri=1Si. This result generalizes to the setting of biforms. Some examples of c...

Definite Determinantal polynomials play a crucial role in semidefinite programming problems. Helton and Vinnikov proved that real zero (RZ) bivariate are definite determinantals. Indeed, general, it is difficult problem to decide whether given polynomial determinantal, if is, of paramount interest determine determinantal representation polynomial. We provide necessary sufficient condition for t...

We observe that the decision problem for the ∃ theory of real closed fields (RCF) is simply reducible to the decision problem for RCF over a connective-free ∀ language in which the only relation symbol is a strict inequality. In particular, every ∃ RCF sentence φ can be settled by deciding a proposition of the form “polynomial p (which is a sum of squares) takes on strictly positive values over...

In this paper Kharitonov's theorem for the robust stability'.df kterval , polynomials is proved using the second method of Lyapunov. The Hermi+' matfi is taken as the matrix of the form which is used as a Lyapunov funct i i tq prove B m i B stability. It is shorn that if the four Hermite matrices eorrespqndmg; to the four Xharitonov extreme pobmomials are positive definite, the Hermitemhtriw of...

In this paper Kharitonov's theorem for the robust stability of interval polynomials is proved using the second method of Lyapunov. The Hermite matrix is taken as the matrix of the quadratic form which is used as a Lyapunov function to prove Hurwitz stability. I t is shown that if the four Hermite matrices corresponding to the four Kharitonov extreme polynomials are positive definite, the Hermit...