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

TL;DR

This paper investigates solutions to specific matrix equations over fields and rings, characterizing their structure, dimension, and conditions for solutions, including triangularizability and uniqueness of solutions in various algebraic settings.

Contribution

It provides new results on the dimension and structure of solution varieties for matrix equations involving nilpotent matrices and polynomial functions, extending understanding over fields and rings.

Findings

Solution variety dimension is n^2-1 for certain equations.

Matrices satisfying the equations are simultaneously triangularizable.

Unique solutions are identified under specific ring and matrix conditions.

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 $AX=XA$.