trace-positive non-commutative polynomials
Ronan Quarez · Proceedings of the American Mathematical Society · 2015
We give some examples of trace-positive non-commutative polynomials of degree $4$ in $3$ variables which are not cyclically equivalent to a sum of hermitian squares. Since some similar examples of degree $6$ in $2$ variables were alreay known, this settles a perfect analogy to Hilbert’s result from the commutative context which says that positive (commutative) polynomials of degree $d$ in $n$ variables are not necessarily sums of squares, the first non-trivial cases being obtained for $(d,n)=(4,3)$ and $(d,n)=(6,2)$.