Abstract
In this paper, via Newton polyhedra, we define and study symmetric matrix polynomials which are nondegenerate at infinity. From this, we construct a class of (not necessarily compact) semialgebraic sets in \(\mathbb {R}^{n}\) such that for each set K in the class, we have the following two statements: (i) the space of symmetric matrix polynomials, whose eigenvalues are bounded on K, is described in terms of the Newton polyhedron corresponding to the generators of K (i.e., the matrix polynomials used to define K) and is generated by a finite set of matrix monomials; and (ii) a matrix version of Schmüdgen’s Positivstellensätz holds: every matrix polynomial, whose eigenvalues are “strictly” positive and bounded on K, is contained in the preordering generated by the generators of K.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
“maximal face” means with respect to the inclusion of faces.
References
Benedetti, R., Risler, J.-J.: Real Algebraic and Semi-algebraic Sets. Hermann, Paris (1990)
Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry, vol. 36. Springer-Verlag, Berlin (1998)
Cimprič, J.: Archimedean operator-theoretic Positivstellensätze. J. Funct. Anal. 260(10), 3132–3145 (2011)
Cimprič, J.: Real algebraic geometry for matrices over commutative rings. J. Algebra 359, 89–103 (2012)
Cimprič, J., Zalar, A.: Moment problems for operator polynomials. J. Math. Anal. Appl. 401(1), 307–316 (2013)
Gindikin, S.G.: Energy estimates connected with the Newton polyhedron. Trudy Moskov. Mat. Obšč 31, 189–236 (1974)
Gindikin, S.G.: Trans. Moscow. Math. Soc. 31, 193–246 (1974)
Hà, H.V., Ho, T.M.: Positive polynomials on nondegenerate basic semi-algebraic sets. Adv. Geom. 16(4), 497–510 (2016)
Hà, H.V., Phạm, T.S.: Genericity in polynomial optimization 3. Series on Optimization and Its Applications. World Scientific Publishing Co. Ltd., Hackensack, NJ (2017)
Klep, I., Schweighofer, M.: Pure states, positive matrix polynomials and sums of hermitian squares. Indiana Univ. Math. J. 59(3), 857–874 (2010)
Kurdyka, K., Michalska, M., Spodzieja, S.: Bifurcation values and stability of algebras of bounded polynomials. Adv. Geom. 14(4), 631–646 (2014)
Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2009)
Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. Emerging Applications of Algebraic Geometry 157–270. IMA Vol. Math. Appl, vol. 149. Springer, New York (2009)
Marshall, M.: Positive polynomials and sum of squares. Mathematical Surveys and Monographs 146. American Mathematical Society, Providence, RI (2008)
Marshall, M.: Polynomials non-negative on a strip. Proc. Am. Math. Soc. 138 (5), 1559–1567 (2010)
Michalska, M.: Algebra of bounded polynomials on a set Zariski closed at infinity cannot be finitely generated. Bull. Sci. Math. 137(6), 705–715 (2013)
Mikhalov, V.P.: The behaviour at infinity of a class of polynomials. Trudy Mat. lnst. Steklov. 91, 59–81 (1967)
Mikhalov, V.P.: Proc. Steklov Inst. Math. 91, 61–82 (1967)
Milnor, J.: Singular Points of Complex Hypersurfaces. Ann. Math. Studies 61, Princeton University Press, Princeton, N.J.: University of Tokyo Press, Tokyo (1968)
Némethi, A., Zaharia, A.: Milnor fibration at infinity. Indag. Math. (N.S.) 3(3), 323–335 (1992)
Nguyen, H., Powers, V.: Polynomials non-negative on strips and half-strips. J. Pure Appl. Algebra 216(10), 2225–2232 (2012)
Phạm, P.P., Phạm, T.S.: Compactness criteria for real algebraic sets and Newton polyhedra. arXiv:1705.10917 (2017)
Plaumann, D., Scheiderer, C.: The ring of bounded polynomials on a semi-algebraic set. Trans. Am. Math. Soc. 364(9), 4663–4682 (2012)
Powers, V.: Positive polynomials and the moment problem for cylinders with compact cross-section. J. Pure Appl. Algebra 188(1–3), 217–226 (2004)
Powers, V., Reznick, B.: Polynomials positive on unbounded rectangles. Positive polynomial in control, 151–163. Lect. Notes Control Inf. Sci, vol. 312. Springer, Berlin (2005)
Prestel, A., Delzell, C.N.: Positive polynomials. From Hilbert’s 17th problem to real algebra. Springer Monographs in Mathematics. Springer-Verlag, Berlin (2001)
Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969–984 (1993)
Scheiderer, C.: Sums of squares of regular functions on real algebraic varieties. Trans. Am. Math. Soc. 352(3), 1039–1069 (1999)
Scheiderer, C.: Sums of squares on real algebraic curves. Math. Z. 245(4), 725–760 (2003)
Scheiderer, C.: Sums of squares on real algebraic surfaces. Manuscripta Math. 119(4), 395–410 (2006)
Scheiderer, C.: Positivity and sums of squares: a guide to recent results. Emerging applications of algebraic geometry, 271–324. IMA Vol. Math. Appl, vol. 149. Springer, New York (2009)
Scherer, C.W., Hol, C.W.J.: Matrix sum-of-squares relaxations for robust semi-definite programs. Math. Program, Ser. B 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.: Global optimization of polynomials using gradient tentacles and sums of squares. SIAM J. Optim. 17(3), 920–942 (2006)
Funding
The first author (Dr. Dinh) is partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED), grant 101.02-2017.310. The second author (Dr. Ho) is partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED), grant 101.04-2017.12. The third author (Dr. Pham) is partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED), grant 101.04-2016.05.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dinh, T.H., Ho, T.M. & Phạm, T.S. A Note on Nondegenerate Matrix Polynomials. Acta Math Vietnam 43, 761–778 (2018). https://doi.org/10.1007/s40306-018-0261-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40306-018-0261-4