Finsler's Lemma for Matrix Polynomials

Finsler’s Lemma charactrizes all pairs of symmetric $n\times n$ real matrices $A$ and $B$ which satisfy the property that $v^{T}Av>0$ for every nonzero $v\in {\mathbb{R}}^n$ such that $v^{T}Bv=0$. 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 $n=1$ reduce to the usual characterizations of positive polynomials on varieties and on compact sets.