Power-central polynomials on matrices

What is it about?

The paper under review is motivated by the following conjecture attributed to Kaplansky: If $p$ is a multilinear polynomial in the free associative algebra over an infinite field $K$, then the set of evaluations of $p$ on the $n\times n$ matrix algebra $M_{n}(K)$ is either $\{0\}$, the field $K$ (viewed as the set of scalar matrices), $\germ{sl}_{n}(K)$, or $M_{n}(K)$. An additional motivation is the well-known fact that the commutator $[x_{1},x_{2}]$ is not a central polynomial for $M_{2}(K)$, but its square $[x_{1},x_{2}]^{2}$ is. The first result shows that if $p$ is neither central nor a polynomial identity, then for $n\geq 4$ its image ${\rm Im}(p)$ in $M_{n}(K)$ contains a matrix of a very specific form. As a consequence, the dimension of the closure in the Zariski topology of ${\rm Im}(p)$ is at least $n^{2}-n+2$. Then the authors study multilinear polynomials $p$ which are $\nu$-central; i.e., $p^{\nu}$ is central and $\nu\geq 1$ is minimal with this property. They show that $\nu$-central polynomials with $\nu>1$ do not exist for $n\geq 4$. For $n=3$ and $\nu>1$, if $\nu$-central polynomials exist, then $\nu=3$, but the problem of their existence is still open. Then, generalizing a result in [D. J. Saltman, Proc. Amer. Math. Soc. 78 (1980), no. 1, 11–13; MR0548073], the authors study $\nu$-central polynomials for $M_{n}({\Bbb Q})$ which are not multilinear and give some restrictions on $\nu$. In particular, they show that 4-central polynomials do not exist. The paper concludes with a couple of open problems. Reviewed by Vesselin Drensky

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