The matrix equations $XA-AX=X^{\alpha}g(X)$ over fields or rings

Abstract

Let $n,\alpha\geq 2$. Let $K$ be an algebraically closed field with characteristic $0$ or greater than $n$. We show that the dimension of the variety of pairs $(A,B)\in {M_n(K)}^2$, with $B$ nilpotent, that satisfy $AB-BA=A^{\alpha}$ or $A^2-2AB+B^2=0$ is $n^2-1$ ; moreover such matrices $(A,B)$ are simultaneously triangularizable. Let $R$ be a reduced ring such that $n!$ is not a zero-divisor and $A$ be a generic matrix over $R$ ; we show that $X=0$ is the sole solution of $AX-XA=X^{\alpha}$. Let $R$ be a commutative ring with unity ; let $A$ be similar to $\mathrm{diag}(\lambda_1I_{n_1},\cdots,\lambda_rI_{n_r})$ such that, for every $i\not= j$, $\lambda_i-\lambda_j$ is not a zero-divisor. If $X$ is a nilpotent solution of $XA-AX=X^{\alpha}g(X)$ where $g$ is a polynomial, then $AX=XA$.