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Acta Mathematica Vietnamica

, Volume 43, Issue 4, pp 761–778 | Cite as

A Note on Nondegenerate Matrix Polynomials

  • Trung Hoa Dinh
  • Toan Minh Ho
  • Tiến Sơn Phạm
Article
  • 95 Downloads

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.

Keywords

Matrix polynomials Positivstellensätze Newton polyhedra Nondegeneracy 

Mathematics Subject Classification (2010)

11E25 13J30 47L07 08B20 41A10 14P10 

Notes

Funding Information

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.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Division of Computational Mathematics and Engineering, Institute for Computational ScienceTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Faculty of Civil EngineeringTon Duc Thang UniversityHo Chi Minh CityVietnam
  3. 3.Department of Mathematics and StatisticsUniversity of North FloridaJacksonvilleUSA
  4. 4.Institute of MathematicsVASTHanoiVietnam
  5. 5.Department of MathematicsUniversity of DalatDalatVietnam

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