Trace-positive non-commutative polynomials

Ronan Quarez (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: Positive semi-definite; Sums of squares; Sums of Hermitian Squares; Trace positivity

Domaines: Géométrie algébrique [math.AG]

Ronan Quarez. Trace-positive non-commutative polynomials. Proceedings of the American Mathematical Society, 2015, 143, pp.3357-3370. ⟨hal-00685397⟩