Positive noncommutative polynomials are sums of squares

• 成果类型: Article
• 署名作者: Helton, JW
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/3597203
• 发表日期: 2002
• 页码: 675-694
• 关键词: algorithm

摘要

Hilbert's 17th problem concerns expression of polynomials on R-n as a sum of squares. It is well known that many positive polynomials are not sums of squares; see [Re], [D'A] for excellent surveys. In this paper we consider symmetric noncommutative polynomials and call one matrix-positive, if whenever matrices of any size are substituted for the variables in the polynomial the matrix value which the polynomial takes is positive semidefinite. The result in this paper is: A polynomial is matrix-positive if and only if it is a sum of squares.