Strict positivstellensätze for matrix polynomials with scalar constraints

We extend Krivine’s strict positivstellensätz for usual (real multivariate) polynomials to symmetric matrix polynomials with scalar constraints. The proof is an elementary computation with Schur complements. Analogous extensions of Schmüdgen’s and Putinar’s strict positivstellensätz were recently proved by Hol and Scherer using methods from optimization theory.