The generalized commutator (Formula presented.) of a list (Formula presented.) of k real (Formula presented.) matrices is defined as a multilinear skew-symmetric function and the linear operator (Formula presented.) on the vector space (Formula presented.) is defined by (Formula presented.). The Amitsur–Levitzki theorem shows that (Formula presented.) when (Formula presented.). We investigate the kernel of T and prove that for all integers k and n such that (Formula presented.) we have (Formula presented.)(Formula presented.)(Formula presented.) where (Formula presented.) if k is even; (Formula presented.) if k is odd and n is even; and (Formula presented.) if k and n are both odd. We conjecture that this result is best possible and that (Formula presented.)(Formula presented.)(Formula presented.) for almost all (Formula presented.) when k and n are in this range. This conjecture is supported by some computational evidence but so far remains open.

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Keywords Amitsur–Levitzki theorem, Generalized commutator, multilinear, skewsymmetric
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Journal Linear and Multilinear Algebra
Dixon, J.D. (John D.), & Pressman, I.S. (Irwin S.). (2017). Generalized commutators and a problem related to the Amitsur–Levitzki theorem. Linear and Multilinear Algebra, 1–9. doi:10.1080/03081087.2017.1389851