Sharp degree bounds for higher-dimensional reduced models

Determine whether there exists a sharp upper bound for the degree of an r-dimensional reduced discrete model M⊆Δ_n parameterized as in Proposition 6.1, expressed in terms of n and r, for every r≥2.

Background

For one-dimensional reduced models, the paper proves the sharp degree bound deg(M)≤2n−1. In higher dimensions, the relevant parameter exponents no longer necessarily yield the total degree of the associated polynomial, so the one-dimensional argument does not directly provide a degree bound for the model itself. The authors therefore pose the existence of a sharp bound depending on the dimension and support size as an explicit question.

References

Let $r\ge2$. For $r$-dimensional reduced models $M\subseteq\Delta_n$ parameterized as in Proposition \ref{prop:model-generalization} does there exist a sharp upper bound for $\deg(M)$ in terms of $n$ and $r$?

— One-dimensional Discrete Models of Maximum Likelihood Degree One  (2507.18686 - Améndola et al., 24 Jul 2025) in Section 6, “Outlook,” Question 6.2