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The L'vov-Kaplansky conjecture states that the image of a multilinearnoncommutative polynomial $f$ in the matrix algebra $M_n(K)$ is a vector spacefor every $n \in {\mathbb N}$. We prove this conjecture for the case where $f$has degree $3$ and $K$ is an algebraically closed field of characteristic $0$.

The L'vov-Kaplansky Conjecture for Polynomials of Degree Three