Abstract
When we move focus from positive semidefinite non-commutative polynomials to trace-positive non-commutative polynomials we naturally meet cyclic equivalence to hermitian squares, see Definitions 1.57 and 1.60 In this chapter we will consider the question whether an nc polynomial is cyclically equivalent to SOHS, i.e., whether it is a member of the cone \(\varTheta ^{2}\), which is a sufficient condition for trace-positivity. A special attention will be given to algorithmic aspects of detecting members in \(\varTheta ^{2}\). We present a tracial version of the Gram matrix method based on the tracial version of the Newton chip method which by using semidefinite programming efficiently answers the question if a given nc polynomial is or is not cyclically equivalent to a sum of hermitian squares.
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Burgdorf, S., Klep, I., Povh, J. (2016). Cyclic Equivalence to Sums of Hermitian Squares. In: Optimization of Polynomials in Non-Commuting Variables. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-33338-0_3
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DOI: https://doi.org/10.1007/978-3-319-33338-0_3
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