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

By Vlad Timofte. 2003

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Abstract We prove that a real symmetric polynomial inequality of degree d⩾2 holds on R + n 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 R n are outlined. The main results are Theorems 5.1 and 5.2 with Corollaries 2.1 and 5.2, and the corresponding results for R n from the last subsection. Part II will contain a discussion on the ordered vector space H d [n] 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).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d659637a0ed003556f2055bc2c208e2fb65ddf7c