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 \(\mathcal {G}\).
References
Cimprič, J.: A representation theorem for Archimedean quadratic modules on \(*\)-rings. Can. Math. Bull. 52(1), 39–52 (2009)
Cimprič, J.: Real algebraic geometry for matrices over commutative rings. J. Algebra 359, 89–103 (2012)
Cimprič, J., Zalar, J.: Moment problems for operator polynomials. J. Math. Anal. App. 401(1), 307–316 (2013)
Gondard, D., Ribenboim, P.: Le 17e problème de Hilbert pour les matrices. Bull. Sci. Math. 98(1), 49–56 (1974)
Jacobi, T., Prestel, A.: Distinguished representations of strictly positive polynomials. J. Reine Angew. Math. 532, 223–235 (2001)
Klep, I., Schweighofer, M.: Pure states, positive matrix polynomials and sums of Hermitian squares. Indiana Univ. Math. J. 59(3), 857–874 (2010)
Krivine, J.-L.: Anneaux préodonnés. J. Anal. Math. 12, 307–326 (1964)
Marshall, M.: Positive Polynomials and Sums of Squares. Mathematical Surveys and Monographs, vol. 146. American Mathematical Society, Providence (2008)
Scheiderer, C.: Sums of squares on real algebraic curves. Math. Z. 245, 725–760 (2003)
Scheiderer, C.: Distinguished representations of non-negative polynomials. J. Algebra 289, 558–573 (2005)
Scherer, C.W., Hol, C.W.J.: Matrix sum-of-squares relaxations for robust semi-definite programs. Math. Progr. Ser. B 107(1–2), 189–211 (2006)
Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Operator Theory: Advances and Applications. Birkhäuser, Basel (1990)
Schmüdgen, K.: A strict Positivstellensatz for the Weyl algebra. Math. Ann. 331, 779–794 (2005)
Schmüdgen, K.: Noncommutative real algebraic geometry: some basic concepts and first ideas. In: Proceedings of Emerging Applications of Algebraic Geometry, IMA Journal of Applied Mathematics, vol. 149, pp. 325–350. Springer, New York (2009)
Schweighofer, M.: Global optimization of polynomials using gradient tentacles and sums of squares. SIAM J. Optim. 17(3), 920–942 (2006)
Stengle, G.: A Nullstellensatz and a Positivstellensatz in semialgebraic geometry. Math. Ann. 207, 87–97 (1974)
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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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
Keywords
- Positive polynomials
- Matrix polynomials
- Sum of squares
- Positivstellensätze
- Local-global principle
- Hessian conditions
- Boundary Hessian conditions