A NOTE ON POSITIVSTELLENSATZE FOR ¨ MATRIX POLYNOMIALS

Abstract

In this paper we give a note on the relation of the positivity of polyno

mial matrices and their homogenizations on basic closed semi-algebraic

sets. Base on this relation, we extend Putinar-Vasilescu’s Positivstel

lensatz, 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. Lˆe, Some Positivstellens¨atze for

polynomial matrices, Positivity 19 (3) (2015), 513-528].