On the positivity of symmetric polynomial functions. I: General results (Q1406527)

Let \(f(X_1,\dots,X_n)\) be a symmetric polynomial of degree \(d\geq 2\). The author studies the problem when \(f(X_1,\dots,X_n)\geq 0\) for all \((X_1,\dots,X_n)\in {\mathbb R}_+^n\). The main result is that this inequality holds if and only if it holds for all elements with at most \(\lfloor d/2\rfloor\) (the integer part) distinct non-zero components (which may have multiplicities). In order to establish this result the author uses that \(f\) can be expressed in terms of the symmetric power sums and solves 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 a restriction of the considered polynomial. The author obtains also analogues of the main results for inequalities which hold for polynomials of even degree and for all \((X_1,\dots,X_n)\in {\mathbb R}^n\).