Positivstellensätze for polynomial matrices
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)]. Similar generalizations to the matrix case are proved for theorems in [\textit{P. J. C. Dickinson} and \textit{J. Povh}, J. Glob. Optim. 61, No. 4, 615--625 (2015; Zbl 1379.11037); \textit{M. Marshall}, Can. Math. Bull. 46, No. 3, 400--418 (2003; Zbl 1080.14069)].
scientific article; zbMATH DE number 7436687
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