trace-positive non-commutative polynomials

Quarez, Ronan

doi:10.1090/S0002-9939-2015-12450-1

trace-positive non-commutative polynomials
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Proc. Amer. Math. Soc. 143 (2015), 3357-3370 Request permission

Abstract:

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 $(d,n)=(6,2)$.

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