Positive symmetric functions
The purpose of this paper is to explain how the classical ideas of Artin-Schreier theory can be used to give various analogues of the Hilbert 17th problem. We give two examples, positive symmetric functions and positive orthogonal invariants. For positive symmetric functions we show that any such function \(f\) has the form: \[ f = \sum_{1<i_1<i_2\le n} \lambda_{i_1 i_2\cdots i_k} \Delta_{i_1}\Delta_{i_2}\cdots \Delta_{i_k} \] where \(\lambda_{i_1 i_2\cdots i_k}\) is a sum of squares of symmetric functions while the \(\Delta_i\) \((i=2,\dots,n)\) are principal minors of the Bezoutian matrix (i.e. the matrix having as \(i,j\) entry the \(i+ j -2\) Newton symmetric function).
scientific article; zbMATH DE number 3595290