Strict Positivstellensätze for matrix polynomials with scalar constraints

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