On the positivity of symmetric polynomial functions.: Part I: General results

We prove that a real symmetric polynomial inequality of degree d?2 holds on if and only if it holds for elements with at most ⌊d/2⌋ distinct non-zero components, which may have multiplicities. We establish this result by solving a Cauchy problem for ordinary differential equations involving the symmetric power sums; this implies the existence of a special kind of paths in the minimizer of some restriction of the considered polynomial function. In the final section, extensions of our results to the whole space are outlined. The main results are Theorems 5.1 and 5.2 with Corollaries 2.1 and 5.2, and the corresponding results for from the last subsection. Part II will contain a discussion on the ordered vector space in general, as well as on the particular cases of degrees d=4 and d=5 (finite test sets for positivity in the homogeneous case and other sufficient criteria).