The Matrix Equation XA- AX=Xαg(X) over Fields or Rings

Abstract

Let n, α ≥ 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) ∈ Mn(K) 2 , with B nilpotent, that satisfy AB − BA = A α 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 α . Let R be a commutative ring with unity ; let A be similar to diag(λ 1 In 1 , · · · , λrIn r ) such that, for every i = j, λ i − λ j is not a zero-divisor. If X is a nilpotent solution of XA − AX = X α g(X) where g is a polynomial, then AX = XA.