Strict Positivstellens\"atze for matrix polynomials with scalar constraints

TL;DR

This paper extends classical positivity certificates to symmetric matrix polynomials with scalar constraints, using elementary Schur complement computations, advancing the theoretical foundation for matrix polynomial positivity.

Contribution

It provides a new elementary proof for strict Positivstellensätze for matrix polynomials, expanding prior scalar results to the matrix setting.

Findings

Extended Krivine's strict positivstellensatz to matrix polynomials

Provided elementary proof using Schur complements

Enhanced theoretical understanding of positivity conditions for matrix polynomials

Abstract

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.