Pfister’s Theorem Fails in the Free Case

Abstract

Artin solved Hilbert’s 17th problem by showing that every positive semidefinite polynomial can be realized as a sum of squares of rational functions. Pfister gave a bound on the number of squares of rational functions: if p is a positive semi-definite polynomial in n variables, then there is a polynomial q so that q 2 p is a sum of at most 2 n squares. As shown by D’Angelo and Lebl, the analog of Pfister’s theorem fails in the case of Hermitian polynomials. Specifically, it was shown that the rank of any multiple of the polynomial \(\|Z\|^{2d} \equiv (\sum_j|z_j|^2)^d\) is bounded below by a quantity depending on d. Here we prove that a similar result holds in a free *-algebra.