Decomposition of a Polynomial as a Sum-of-Squares of Polynomials and the S-Procedure

This paper investigates links between the problem of determining a decomposition of a polynomial as a sum-of-squares of polynomials and the S-Procedure. We first show that the S-Procedure can be used to check whether a given polynomial is non-negative. Then, using mostly linear algebra arguments, we show that this non-negativity test leads to an affirmative answer if and only if such polynomial admits a decomposition as a sum-of-squares of polynomials.