Commuting Nilpotent Operators and Maximal Rank

Let X, (X) over tilde be commuting nilpotent matrices over k with nilpotency p(t), where k is an algebraically closed field of positive characteristic p. We show that if X - (X) over tilde is a certain linear combination of products of pairwise commuting nilpotent matrices, then X is of maximal rank if and only if (X) over tilde is of maximal rank.