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Positivstellensätze for polynomial matrices

Abstract

In this paper we establish some Positivstellensätze for polynomial matrices, applying the Scherer–Hol theorem. Firstly, we give a representation for polynomial matrices positive definite on subsets of compact polyhedra. Then we establish a Putinar-Vasilescu Positivstellensatz for polynomial matrices. Next we propose a matrix version of the Dickinson–Povh Positivstellensatz. Finally, we establish a version of Marshall’s theorem for polynomial matrices, approximating positive semi-definite polynomial matrices using sums of squares.

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Acknowledgements

The authors would like to express their sincere gratitude to Prof. Konrad Schmüdgen for fruitful discussions on representation theory for the algebra of matrices. They would also like to thank the anonymous referees for their useful comments and suggestions. This paper was finished during the visit of the second and the third authors at the Vietnam Institute for Advanced Study in Mathematics (VIASM). They thanks VIASM for financial support and hospitality. This research was funded by the Vietnam Ministry of Education and Training under the decision number 3813/QĐ-BGDĐT dated November 20, 2020.

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Correspondence to Cong Trinh Le.

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Dinh, T.H., Ho, M.T. & Le, C.T. Positivstellensätze for polynomial matrices. Positivity 25, 1295–1312 (2021). https://doi.org/10.1007/s11117-021-00816-7

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  • DOI: https://doi.org/10.1007/s11117-021-00816-7

Keywords

  • Polynomial matrix
  • Scherer–Hol’s theorem
  • Positivstellensatz
  • Pólya’s theorem
  • Putinar–Vasilescu’s theorem
  • Marshall’s theorem

Mathematics Subject Classification

  • 15A48
  • 15A54
  • 11E25
  • 13J30
  • 14P10