A Note on Nondegenerate Matrix Polynomials

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. 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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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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Keywords

  • Matrix polynomials
  • Positivstellensätze
  • Newton polyhedra
  • Nondegeneracy

Mathematics Subject Classification (2010)

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