Some Positivstellensätze for polynomial matrices
arXiv:1403.3783 · doi:10.1007/s11117-014-0312-6
Abstract
In this paper we give a version of Krivine-Stengle's Positivstellensatz, Schweighofer's Positivstellensatz, Scheiderer's local-global principle, Scheiderer's Hessian criterion and Marshall's boundary Hessian conditions for polynomial matrices, i.e. matrices with entries from the ring of polynomials in the variables (x_1,...,x_d) with real coefficients. Moreover, we characterize Archimedean quadratic modules of polynomial matrices, and study the relationship between the compactness of a subset in (\r^{d}) with respect to a subset (\hoa{G}) of polynomial matrices and the Archimedean property of the preordering and the quadratic module generated by (\hoa{G}).
14 pages, Section 5 rewritten, three references added, final version, to be published in Positivity