A NOTE ON POSITIVSTELLENS¨ATZE FOR MATRIX POLYNOMIALS

Thi H`oa-B`ình Du

Thi H`oa-B`ình Du

Keywords: P´olya’s theorem, Putinar-Vasilescu’s theorem, Reznick’s theorem, Polynomial matrix

Abstract

In this paper we give a note on the relation of the positivity of polynomial matrices and their homogenizations on basic closed semi-algebraic sets. Base on this relation, we extend Putinar-Vasilescu’s Positivstellensatz, in particular, Reznick’s Positivstellensatz, and Dickinson-Povh’s Positivstellensatz to (not necessarily homogeneous) polynomial matrices. This is a continuation of the work [C.-T. Lˆ e, Some Positivstellens¨atze for polynomial matrices, Positivity 19 (3) (2015), 513-528].