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

Some Positivstellensätze for polynomial matrices

  • Lê Công-Trình


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 \(\mathcal {G}\).


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

Mathematics Subject Classfication

14P99 13J30 15B33 15B48 



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.


  1. 1.
    Cimprič, J.: A representation theorem for Archimedean quadratic modules on \(*\)-rings. Can. Math. Bull. 52(1), 39–52 (2009)Google Scholar
  2. 2.
    Cimprič, J.: Real algebraic geometry for matrices over commutative rings. J. Algebra 359, 89–103 (2012)Google Scholar
  3. 3.
    Cimprič, J., Zalar, J.: Moment problems for operator polynomials. J. Math. Anal. App. 401(1), 307–316 (2013)Google Scholar
  4. 4.
    Gondard, D., Ribenboim, P.: Le 17e problème de Hilbert pour les matrices. Bull. Sci. Math. 98(1), 49–56 (1974)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Jacobi, T., Prestel, A.: Distinguished representations of strictly positive polynomials. J. Reine Angew. Math. 532, 223–235 (2001)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Klep, I., Schweighofer, M.: Pure states, positive matrix polynomials and sums of Hermitian squares. Indiana Univ. Math. J. 59(3), 857–874 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Krivine, J.-L.: Anneaux préodonnés. J. Anal. Math. 12, 307–326 (1964)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Marshall, M.: Positive Polynomials and Sums of Squares. Mathematical Surveys and Monographs, vol. 146. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar
  9. 9.
    Scheiderer, C.: Sums of squares on real algebraic curves. Math. Z. 245, 725–760 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Scheiderer, C.: Distinguished representations of non-negative polynomials. J. Algebra 289, 558–573 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    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)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Operator Theory: Advances and Applications. Birkhäuser, Basel (1990)CrossRefGoogle Scholar
  13. 13.
    Schmüdgen, K.: A strict Positivstellensatz for the Weyl algebra. Math. Ann. 331, 779–794 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    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)Google Scholar
  15. 15.
    Schweighofer, M.: Global optimization of polynomials using gradient tentacles and sums of squares. SIAM J. Optim. 17(3), 920–942 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Stengle, G.: A Nullstellensatz and a Positivstellensatz in semialgebraic geometry. Math. Ann. 207, 87–97 (1974)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsQuy Nhon UniversityQuy NhonVietnam

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