Handelman's Positivstellensatz for polynomial matrices positive definite on polyhedra

TL;DR

This paper extends Handelman's Positivstellensatz to polynomial matrices that are positive definite on convex polyhedra and provides a method to find such representations, also relating it to Schmüdgen's theorem.

Contribution

It introduces a matrix version of Handelman's Positivstellensatz and a procedure to compute the representation for polynomial matrices positive definite on polyhedra.

Findings

Matrix Handelman's Positivstellensatz established

Procedure for finding polynomial matrix representations proposed

Special case of Schmüdgen's Positivstellensatz derived

Abstract

In this paper we give a matrix version of Handelman's Positivstellensatz [1], representing polynomial matrices which are positive definite on convex, compact polyhedra. Moreover, we propose also a procedure to find such a representation. As a corollary of Handelman's theorem, we give a special case of Schm\"udgen's Positivstellensatz for polynomial matrices positive definite on convex, compact polyhedra.