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

✍ Scribed by Vlad Timofte

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 177 KB
• Volume: 284
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1016/s0022-247x(03)00301-9

✦ Synopsis

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 [n] d 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).