Positivstellensätze for polynomial matrices
type: Item
instance of: scholarly article
title: Positivstellensätze for polynomial matrices
zbMATH Open document ID: 1481.15021
author: Dinh Trung Hoa
author: Minh Toan Ho
author: Lê Công-Trình
published in: Positivity
publication date: 2021-12-01T00:00:00Z
review text: Hilbert's 17th problem (i.e., ``a real polynomial in \(n\) variables which takes non-negative values on all of \(\mathbb{R}^{n}\) can be written as a sum of squares of real rational functions'') and variations of it have been extensively studied in the literature. In particular, similar theorems have been proved for matrices with polynomial entries. The authors of this paper use a theorem of \textit{C. W. Scherer} and \textit{C. W. J. Hol} [Math. Program. 107, No. 1--2 (B), 189--211 (2006; Zbl 1134.90033)] to prove some matrix Positivstellensätze in a unified way. Let \(\mathcal{M}_{t}\) be the ring of \(t\times t\) matrices over the ring \( \mathbb{R}[X]\) of real polynomials in \(n\) variables \(X_{1},\dots,X_{n}\), and let \(\mathcal{S}_{t}\) be the subset of symmetric matrices. A subset \( \mathcal{Q}\) of \(\mathcal{S}_{t}\) is called a quadratic module if: \(\mathbf{I }\in \mathcal{Q}\); \(\mathcal{Q+Q\subseteq Q}\); and \(\mathbf{A}^{T}\mathcal{Q} \mathbf{A\subseteq }\mathcal{Q}\) for all \(\mathbf{A\in }\mathcal{M}_{t}.\) If \(\mathcal{G\subseteq S}_{t}\) then \(\mathcal{M(G})\) is the smallest quadratic module containing \(\mathcal{G}\) and \(K(\mathcal{G})\) is the set of all \(x\in \mathbb{R}^{n}\) such that \(\mathbf{G}(x)\) is positive semidefinite for all \( \mathbf{G}\in \mathcal{G}\). One example of the theorems proved in this paper is the following. Theorem. Let \(\mathcal{G}\subseteq \mathcal{S}_{t}\) be a finite set of homogeneouos polynomial matrices of even degrees, and let \( \mathbf{F\in }\mathcal{S}_{t}\) be a homogeneous polynomial matrix of even degree \(>0\). If \(\mathbf{F(}x\mathbf{)}\) is positive definite for all nonzero \(x\) in \(K(\mathcal{G})\), then for some integer \(N>0\) we have \[ \left( \sum_{i=1}^{n}X_{i}^{2}\right) ^{N}\mathbf{F\in }\mathcal{M}(G). \] This is a matrix version of the (scalar) Theorem 4.5 in [\textit{M. Putinar} and \textit{F.-H. Vasilescu}, Ann. Math. (2) 149, No. 3, 1087--1107 (1999; Zbl 0939.44003)]. Simil…
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 13J30
Mathematics Subject Classification ID: 14P10
zbMATH DE Number: 7436687
zbMATH Keywords: polynomial matrix
zbMATH Keywords: Scherer-Hol theorem
zbMATH Keywords: Positivstellensätze
zbMATH Keywords: Pólya theorem
zbMATH Keywords: Putinar-Vasilescu theorem
zbMATH Keywords: Marshall theorem
MaRDI profile type: Publication
full work available at URL: s11117-021-00816-7
OpenAlex ID: W3126318862
cites work: A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings
cites work: Real algebraic geometry for matrices over commutative rings
cites work: Moment problems for operator polynomials
cites work: On an extension of Pólya's Positivstellensatz
cites work: Q4686053
cites work: Representing polynomials by positive linear functions on compact convex polyhedra
cites work: Pure states, positive matrix polynomials and sums of hermitian squares
cites work: Anneaux preordonnes
cites work: Q5730197
cites work: Some positivstellensätze for polynomial matrices
cites work: Handelman's Positivstellensatz for polynomial matrices positive definite on polyhedra
cites work: Approximating Positive Polynomials Using Sums of Squares
cites work: Q4285035
cites work: Solving moment problems by dimensional extension
cites work: Sums of squares on real algebraic curves
cites work: Distinguished representations of non-negative polynomials
cites work: Matrix sum-of-squares relaxations for robust semi-definite programs
cites work: The \(K\)-moment problem for compact semi-algebraic sets
cites work: A strict Positivstellensatz for the Weyl algebra
cites work: Non-commutative Real Algebraic Geometry - Some Basic Concepts and First Ideas
cites work: The Moment Problem
cites work: Global Optimization of Polynomials Using Gradient Tentacles and Sums of Squares
cites work: A Nullstellensatz and a Positivstellensatz in semialgebraic geometry
DOI: 10.1007/S11117-021-00816-7
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description: scientific article; zbMATH DE number 7436687