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Finsler's Lemma charactrizes all pairs of symmetric $n \times n$ real matrices $A$ and $B$ which satisfy the property that $v^T A v>0$ for every nonzero $v \in \mathbb{R}^n$ such that $v^T B v=0$. 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., 23 pages, 2 figures, submitted

We extend Krivine’s strict positivstellensatz for usual (real multivariate) polynomials to symmetric matrix polynomials with scalar constraints. The proof is an elementary computation with Schur complements. Analogous extensions of Schmudgen’s and Putinar’s strict positivstellensatz were recently proved by Hol and Scherer using methods from optimization theory.

The n -th product level of a skew–field D , ps n ( D ) , is a generalization of the n -th level of a field F , s n ( F ) . An explicit bound for s 2 m ( F ) in terms of m and s 2 ( F ) is known and it is also known that there is no such bound for ps 2 m ( D ) when m is even. Our aim is to explicitly construct such a bound for odd m . More precisely, we construct a function f : N 3 → N , such that ps 2 k l ( D ) ⩽ f ( ps 2 k ( D ) , k , l ) , for every integer k , every odd integer l and every skew–field D . We give an explicit bound for the n -th Pythagorean number of a simple extension Q ( d ) / Q in Section 2 and prove a noncommutative version of Hilbert identities in Section 3. These two results are used in Section 4 in the proof of our main result. In section 5, we show that the n -th product level of an Ore domain is equal to the n -th product level of its skew field of fractions.