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 Schmudgen’s Positivstellensatz holds: every matrix polynomial, whose eigenvalues are “strictly” positive and bounded on K, is contained in the preordering generated by the generators of K.