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A Note on Nondegenerate Matrix Polynomials

A Note on Nondegenerate Matrix Polynomials

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 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*.