Positivity

, Volume 22, Issue 2, pp 449–460 | Cite as

Handelman’s Positivstellensatz for polynomial matrices positive definite on polyhedra

Article

Abstract

In this paper we give a matrix version of Handelman’s Positivstellensatz (Handelman in Pac J Math 132:35–62, 1988), representing polynomial matrices which are positive definite on convex, compact polyhedra. Moreover, we propose also a procedure to find such a representation. As a corollary of Handelman’s theorem, we give a special case of Schmüdgen’s Positivstellensatz for polynomial matrices positive definite on convex, compact polyhedra.

Keywords

Handelman’s theorem Pólya’s theorem Schmüdgen’s theorem Matrix polynomial Polynomial matrix Positivstellensatz Positive definite Standard simplex Polyhedron 

Mathematics Subject Classification

14P99 14Q99 14P10 52B99 15B48 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their useful comments and suggestions. They would also like to thank Dr. Ngo Lam Xuan Chau for his fruitful discussion to compute the kernel of the homomorphism \(M_\varphi \). Both authors are partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2016.27.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsQuy Nhon UniversityQuy NhonVietnam

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