Shengyong Pan

math.RA — Rings and Algebras

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

Contributed by Shengyong Pan

Let $K$ 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 $n\geq3$, a nonidentity has image $\mathfrak{sl}_n(K)$ when its cyclic class is zero and $M_n(K)$ otherwise. In these degrees the only nonzero identity exception is the standard degree-six line on $M_3(K)$. 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.