Trace-positive non-commutative polynomials. (English) Zbl 1396.16017
Summary: 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 \((d,n)=(6,2)\).
MSC:
| 16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |
| 15A63 | Quadratic and bilinear forms, inner products |
Software:
NCSOStoolsReferences:
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