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Positivity

, Volume 19, Issue 3, pp 513–528 | Cite as

Some Positivstellensätze for polynomial matrices

  • Lê Công-Trình
Article

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 \(x_1,\ldots ,x_d\) with real coefficients. Moreover, we characterize Archimedean quadratic modules of polynomial matrices, and study the relationship between the compactness of a subset in \(\mathbb R^{d}\) with respect to a subset \(\mathcal {G}\) of polynomial matrices and the Archimedean property of the preordering and the quadratic module generated by \(\mathcal {G}\).

Keywords

Positive polynomials Matrix polynomials Sum of squares Positivstellensätze Local-global principle  Hessian conditions  Boundary Hessian conditions 

Mathematics Subject Classfication

14P99 13J30 15B33 15B48 

Notes

Acknowledgments

The author would like to thank the anonymous referees for their useful comments and suggestions. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant number 101.99-2013.24. This work is finished during the author’s postdoctoral fellowship at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He thanks VIASM for financial support and hospitality.

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsQuy Nhon UniversityQuy NhonVietnam

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