Optimization of Polynomials in Non-Commuting Variables

Abstract

In Chap. 3 trace-positivity together with the question how to detect it was explored in details. Due to hardness of the decision problem “Is a given nc polynomial f trace-positive?” we proposed a relaxation of the problem, i.e., we are asking if f is cyclically equivalent to SOHS. The tracial Gram matrix method based on the tracial Newton polytope was proposed (see Sects. 3.3 and 3.4) to efficiently detect such polynomials.