# Some Positivstellensätze for polynomial matrices

Some Positivstellensätze for polynomial matrices In this paper we give a version of Krivine–Stengle’s Positivstellensatz, Schweighofer’s Positivstellensatz, Scheiderer’s local-global principle, Scheiderer’s Hessian criterion and Marshall’s boundary Hessian conditions for polynomial matrices, i.e. matrices with entries from the ring of polynomials in the variables $$x_1,\ldots ,x_d$$ x 1 , … , x d with real coefficients. Moreover, we characterize Archimedean quadratic modules of polynomial matrices, and study the relationship between the compactness of a subset in $$\mathbb R^{d}$$ R d with respect to a subset $$\mathcal {G}$$ G of polynomial matrices and the Archimedean property of the preordering and the quadratic module generated by $$\mathcal {G}$$ G . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Some Positivstellensätze for polynomial matrices

, Volume 19 (3) – Oct 22, 2014
16 pages

/lp/springer_journal/some-positivstellens-tze-for-polynomial-matrices-z1CRac68jO
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0312-6
Publisher site
See Article on Publisher Site

### Abstract

In this paper we give a version of Krivine–Stengle’s Positivstellensatz, Schweighofer’s Positivstellensatz, Scheiderer’s local-global principle, Scheiderer’s Hessian criterion and Marshall’s boundary Hessian conditions for polynomial matrices, i.e. matrices with entries from the ring of polynomials in the variables $$x_1,\ldots ,x_d$$ x 1 , … , x d with real coefficients. Moreover, we characterize Archimedean quadratic modules of polynomial matrices, and study the relationship between the compactness of a subset in $$\mathbb R^{d}$$ R d with respect to a subset $$\mathcal {G}$$ G of polynomial matrices and the Archimedean property of the preordering and the quadratic module generated by $$\mathcal {G}$$ G .

### Journal

PositivitySpringer Journals

Published: Oct 22, 2014

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