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A Note on Nondegenerate Matrix Polynomials

Acta Mathematica Vietnamica volume 43, pages 761–778 (2018)Cite this article

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.

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Notes

  1. “maximal face” means with respect to the inclusion of faces.

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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.

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Authors and Affiliations

  1. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Trung Hoa Dinh

  2. Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Trung Hoa Dinh

  3. Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL, 32224, USA

    Trung Hoa Dinh

  4. Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam

    Toan Minh Ho

  5. Department of Mathematics, University of Dalat, 1 Phu Dong Thien Vuong, Dalat, Vietnam

    Tiến Sơn Phạm

Authors
  1. Trung Hoa Dinh
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  2. Toan Minh Ho
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  3. Tiến Sơn Phạm
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Correspondence to Toan Minh Ho.

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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

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  • DOI: https://doi.org/10.1007/s40306-018-0261-4

Keywords

  • Matrix polynomials
  • Positivstellensätze
  • Newton polyhedra
  • Nondegeneracy

Mathematics Subject Classification (2010)

  • 11E25
  • 13J30
  • 47L07
  • 08B20
  • 41A10
  • 14P10