Pfister's theorem fails in the free case

TL;DR

This paper investigates the limitations of Pfister's theorem in the context of free *-algebras, showing that a similar lower bound on polynomial rank applies in this non-commutative setting.

Contribution

It extends the failure of Pfister's theorem from Hermitian polynomials to free *-algebras, establishing a lower bound on polynomial rank in this non-commutative framework.

Findings

Pfister's theorem does not hold in free *-algebras.

A lower bound on polynomial rank similar to the Hermitian case is proven.

The result generalizes the failure of Pfister's theorem to a broader algebraic setting.

Abstract

Artin solved Hilbert's $17^{th}$ 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 $q^2p$ 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 $\ast$-algebra.