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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cimpric̆, J.: Strict Positivstellensätze for matrix polynomials with scalar constraints. Linear Algebra Appl. 434(8), 1879–1883 (2011)
Cimpric̆, J.: Real algebraic geometry for matrices over commutative rings. J. Algebra 359, 89–103 (2012)
Cimpric̆, J., Zalar, A.: Moment problems for operator polynomials. J. Math. Anal. Appl. 401(1), 307–316 (2013)
Handelman, D.: Representing polynomials by positive linear functions on compact convex polyhedra. Pac. J. Math. 132, 35–62 (1988)
Henrion, D., Lasserre, J., Loefberg, J.: GloptiPoly 3: Moments, Optimization and Semidefinite Programming. http://homepages.laas.fr/henrion/software/gloptipoly3/
Krivine, J.L.: Quelques propértiés des préordres dans les anneaux commutatifs unitaires. C. R. Acad. Sci. Paris 258, 3417–3418 (1964)
Lê, C.-T.: Some Positivstellensätze for polynomial matrices. Positivity 19(3), 513–528 (2015)
Marshall, M.: Positive Polynomials and Sums of Squares, Mathematical Surveys and Monographs. American Mathematical Society, Providence (2008)
Pólya, G.: Über positive Darstellung von Polynomen Vierteljschr. Naturforsch. Ges. Zürich 73, 141–145 (1928). In: Boas, R.P. (ed.) Collected Papers, vol. 2, pp. 309–313. MIT Press, Cambridge (1974)
Powers, V., Reznick, B.: A new bound for Pólya’s theorem with applications to polynomials positive on polyhedra. J. Pure Appl. Algebra 164, 221–229 (2001)
Prestel, A., Delzell, C.N.: Positive Polynomials—From Hilbert’s 17th Problem to Real Algebra. Springer, New York (2001)
Savchuk, Y., Schmüdgen, K.: Positivstellensätze for algebras of matrices. Linear Algebra Appl. 43, 758–788 (2012)
Scherer, C.W., Hol, C.W.J.: Matrix sum-of-squares relaxations for robust semi-definite programs. Math. Program. 107(1–2), 189–211 (2006)
Schmüdgen, K.: The K-moment problem for compact semi-algebraic sets. Math. Ann. 289(2), 203–206 (1991)
Schweighofer, M.: An algorithmic approach to Schmüdgen’s Positivstellensatz. J. Pure Appl. Algebra 166(3), 307–319 (2002)
Zedek, M.: Continuity and location of zeros of linear combinations of polynomials. Proc. Am. Math. Soc. 16, 78–84 (1965)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lê, CT., Dư, THB. Handelman’s Positivstellensatz for polynomial matrices positive definite on polyhedra. Positivity 22, 449–460 (2018). https://doi.org/10.1007/s11117-017-0520-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-017-0520-y
Keywords
- Handelman’s theorem
- Pólya’s theorem
- Schmüdgen’s theorem
- Matrix polynomial
- Polynomial matrix
- Positivstellensatz
- Positive definite
- Standard simplex
- Polyhedron