Degree Bounds for Polynomial Verification of the Matrix Cube Problem

Abstract

In this paper we consider the problem of how to computationally test whether a matrix inequality is positive semidefinite on a semialgebraic set. We propose a family of sufficient conditions using the theory of matrix Positivstellensatz refutations. When the semialgebraic set is a hypercube, we give bounds on the degree of the required certificate polynomials.