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Upper bounds for N_n(F) when finite

Establish upper bounds for N_n(F) in cases where N_n(F) is finite, with particular attention to the setting where F is a finite field.

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Background

The paper proves a general lower bound N_n ≥ C_{n−1} and shows this bound is strict for n ≥ 4. While lower bounds are available, the authors point out that determining upper bounds remains open, especially in finite-field contexts where N_n(F) is known to be finite.

They explicitly leave the problem of finding upper bounds for future work.

References

On the other side, another problem is to find upper bounds for N_n( ), when this number is finite, in particular, when is a finite field. We do not address this problem in this article, but we left it open for future works.

— Invariants for isomorphism classes in the category $\bcalNT$ (2508.00084 - Maturana, 31 Jul 2025) in Introduction (Section 1)