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A Note on Nondegenerate Matrix Polynomials
Trung Hoa Dinh
,
Toan Minh Ho
,
Tiến Sơn Phạm
Abstract
n 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 $K$.