Finsler's Lemma for matrix polynomials

Finsler's Lemma characterizes all pairs of symmetric n×nn×n real matrices A and B   which satisfy the property that vTAv>0vTAv>0 for every nonzero v∈Rnv∈Rn such that vTBv=0vTBv=0. We extend this characterization to all symmetric matrices of real multivariate polynomials, but we need an additional assumption that B   is negative semidefinite outside some ball. We also give two applications of this result to Noncommutative Real Algebraic Geometry which for n=1n=1 reduce to the usual characterizations of positive polynomials on varieties and on compact sets.