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

The L’vov-Kaplansky conjecture states that the image of a multilinear noncommutative polynomial $f$ in the matrix algebra $M_n(K)$ is a vector space for 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 $0$.