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The L'vov--Kaplansky conjecture for multilinear polynomials of degrees four to six

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Contributed by Shengyong Pan ↗

Version 1 / Nov 07, 2026 / CC BY 4.0

Abstract

Let KK be an algebraically closed field of characteristic zero. We determine the images of ordinary multilinear polynomials of degrees four, five and six on matrix algebras. For n≥3n\geq3, a nonidentity has image sln(K)\mathfrak{sl}_n(K) when its cyclic class is zero and Mn(K)M_n(K) otherwise. In these degrees the only nonzero identity exception is the standard degree-six line on M3(K)M_3(K). The proof starts with proper sextics and treats arbitrary sextics by unit restrictions and local trace lifting. An image-preserving product substitution then gives degrees four and five as consequences. We also give cyclic and matrix lifting results applicable in arbitrary degree. The finite calculations are given by explicit matrix-unit identities.

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Version history

  1. v1 · 96 pagesSubmitted by Shengyong PanInitial depositCurrentNov 07, 2026