Construct large non-nilpotent matrix spaces over finite fields

Construct, for every finite field F and every integer n e 1, a matrix space W e F^{nn} of dimension nb2-e nb2 such that no nonzero matrix in W is nilpotent, thereby enabling a duality-based proof of the GerstenhaberSerezhkin theorem over all fields.

Background

The paper develops a duality method for extremal problems on matrix spaces. To prove an upper bound on the dimension of spaces consisting entirely of matrices with a specified property, the method seeks a complementary space of the dimension deficit whose nonzero elements avoid that property. The authors apply this strategy successfully to singular matrices and to matrices of bounded rank over all fields by first constructing suitable complementary spaces over finite fields and then using a general reduction theorem.

The GerstenhaberSerezhkin theorem states that a subspace of nn matrices consisting entirely of nilpotent matrices has dimension at most binom(n,2). A complementary construction would require a space of dimension nb2-e nb2 containing no nonzero nilpotent matrix. The paper notes that symmetric matrices provide such a construction over the real numbers, but that the corresponding construction over finite fields is unresolved. The stated conjecture would, through the papers reduction theorem, imply the GerstenhaberSerezhkin dimension bound over every field.

References

It would again be very interesting to find a proof of this theorem using duality, i.e. by exhibiting a space of dimension $n2-\binom n2$ none of whose non-zero elements is nilpotent. This is straightforward over $R$, as the space of symmetric matrices does the job, but it is not clear to us how to do it (or whether it is even true) over finite fields. \begin{conjecture} For every finite field $F$ and every integer $n \geq 1$, there is a matrix space $W \leq F{n\times n}$ with $\dim W = n2-\binom n2$ such that no non-zero matrix in $W$ is nilpotent. \end{conjecture}

— Duality for matrix space questions  (2609.29177 - Wigderson, 24 Sep 2026) in Section 1, immediately following Theorem 3 (the conjecture environment); see also the preceding paragraph discussing the GerstenhaberSerezhkin theorem