Affine spaces of nilpotent and square-zero matrices with constant rank

Over a field with more than three elements, the largest dimension of an affine space of nilpotent matrices of size n≥3 and constant rank two is proved to be $2n-5$. This establishes the rank-two case of a conjecture of Calamai and Rubei. The corresponding maximum for square-zero matrices is also determined.For square-zero matrices of rank r in size $2r$, the maximum is 2 r²/4 over every algebraically closed field, in arbitrary characteristic. The proof uses an exact reduction, valid over fields with more than two elements, to pairs of matrix spaces whose components commute elementwise, followed by a Borel fixed point argument. The pair bound also holds when the commutant of the first component has split semisimple quotient. Over every field with more than two elements, the affine formula is established for r≤6, and the extremal affine space is unique up to similarity when $r=4$. Three nonsimilar extremal families are constructed in every odd rank at least three over algebraically closed fields.Finally, the regular nilpotent case is revisited. The exact field-size thresholds are determined in sizes two and three. In size four, the maximum dimension is proved to be three over F₄ and either three or four over F₃, with structural restrictions on any four-dimensional example.