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A Note on Nondegenerate Matrix Polynomials

Authors: 

Trung Hoa Dinh, Toan Minh Ho, Tiến Son Pham

Source title: 
Acta Mathematica Vietnamica, 43(4): 761-778, 2018 (Scopus)
Academic year of acceptance: 
2018-2019
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   Rn  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.