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Some Positivstellensätze for polynomial matrices

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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 \(\mathcal {G}\).

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Notes

  1. In [12, 13], the term m-admissible wedge was used.

  2. In this case, the ring \(\mathcal {M}_n({\mathbb {R}}[X])\) is called algebraically bounded with respect to the quadratic module \(\mathcal {M}\), cf. [13].

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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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Correspondence to Lê Công-Trình.

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Công-Trình, L. Some Positivstellensätze for polynomial matrices. Positivity 19, 513–528 (2015). https://doi.org/10.1007/s11117-014-0312-6

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  • DOI: https://doi.org/10.1007/s11117-014-0312-6

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