Symmetric semi-algebraic sets and non-negativity of symmetric polynomials

The question of how to certify the non-negativity of a polynomial function lies at the heart of Real Algebra and it also has important applications to Optimization. In the setting of symmetric polynomials Timofte provided a useful way of certifying non-negativity of symmetric polynomials that are of a fixed degree. In this note we present more general results which naturally generalize Timofte's setting. We investigate families of polynomials that allow special representations in terms of power-sum polynomials.These in particular also include the case of symmetric polynomials of fixed degree. Therefore, we recover the consequences of Timofte's original statements as a corollary. Thus, this note also provides an alternative and simple proof of Timofte's original statements.