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 has important applications to optimization. Building on work by Choi, Lam, and Reznick [4], as well as Harris [5], Timofte [9] provided a remarkable method to efficiently certify non-negativity of symmetric polynomials. In this note we slightly generalize Timofte's statements and investigate families of polynomials that allow special representations in terms of power-sum polynomials. We also recover the consequences of Timofte's original statements as a corollary.