Positive symmetric functions
type: Item
instance of: scholarly article
title: Positive symmetric functions
zbMATH Open document ID: 0383.12013
DOI: 10.1016/0001-8708(78)90011-7
published in: Advances in Mathematics
publication date: 1978-01-01T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 12E10
Mathematics Subject Classification ID: 12D99
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 3595290
author: Claudio Procesi
MaRDI profile type: Publication
full work available at URL: 0001-8708(78)90011-7
OpenAlex ID: W2011695558
cites work: On Azumaya algebras and finite dimensional representations of rings
cites work: A characteristic free approach to invariant theory
cites work: Q5328221
cites work: Q5642570
cites work: The invariant theory of \(n\times n\) matrices
cites work: A non-commutative real Nullstellensatz and Hilbert's 17th problem
cites work: Introduction to model theory and to the metamathematics of algebra
description: scientific article; zbMATH DE number 3595290