Abstract
We give some examples of trace-positive non-commutative quaternary quartics which are not cyclically equivalent to a sum of hermitian squares. Since some similar examples of ternary sextics were already known, this settles a perfect analogy to Hilbert's results from the commutative context which says that in general, positive (commutative) polynomials are not necessarily sums of squares, the first non trivial cases being obtained for ternary sextics and quaternary quartics.
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