# Some applications of Scherer-Hol's theorem for polynomial matrices

## TL;DR

This paper explores applications of Scherer-Hol's theorem for polynomial matrices, including positivity representations, Positivstellensatz extensions, and approximation methods for positive semi-definite polynomial matrices.

## Contribution

It introduces new representations and Positivstellensatz results for polynomial matrices, extending existing theorems and proposing approximation techniques.

## Key findings

- Representation for positive definite polynomial matrices on polyhedra
- A matrix version of the Pólya-Putinar-Vasilescu Positivstellensatz
- Approximation of positive semi-definite polynomial matrices using sums of squares

## Abstract

In this paper we establish some applications of the Scherer-Hol's theorem for polynomial matrices. Firstly, we give a representation for polynomial matrices positive definite on subsets of compact polyhedra. Then we establish a Putinar-Vasilescu Positivstellensatz for homogeneous and non-homogeneous polynomial matrices. Next we propose a matrix version of the P\'olya-Putinar-Vasilescu Positivstellensatz. Finally, we approximate positive semi-definite polynomial matrices using sums of squares.