Height inequality for unions of structured submatrices

Determine whether, for subsets I_1 J_1,\ldots,I_s J_s satisfying condition (19) of Theorem 3.8 and T=\bigcup_{\alpha=1}^s I_\alpha J_\alpha, the inequality \operatorname{height}(I_T)\geq\sum_{\alpha=1}^s\operatorname{height}(I_{I_\alpha J_\alpha}) holds, thereby potentially providing an alternative proof of Theorem 3.8 via the general height criterion.

Background

The paper develops two methods for detecting dependent sets in the determinantal matroid: a general height criterion and a construction based on unions of submatrices satisfying the overlap condition in Theorem 3.8. In the examples considered, the height of the determinantal ideal associated with the union agrees with, or exceeds, the sum of the corresponding heights of the individual submatrices.

The authors exhibit an example where the height is strictly larger than that sum, and ask whether the displayed lower bound nevertheless holds under the stated overlap condition. An affirmative answer would show that the general height criterion yields an alternative proof of the dependent-set construction theorem.

References

This raises the following question. Suppose $I_1 J_1, \ldots, I_s J_s$ satisfies condition eq:condition_dep_Asche of Theorem \ref{thm:dependent_Asche}, and let $T= \bigcup_{\alpha=1}s I_\alpha J_\alpha$. Is it true that \height(I_T) \ge \sum_{\alpha=1}s \height(I_{I_\alpha J_\alpha}) \ ?

eq:condition_dep_Asche:

(IαJα)∩(⋃k<αIkJk)⊆Iα′Jα′where min⁡(∣Iα′∣,∣Jα′∣)≤r (I_\alpha J_\alpha) \cap \biggl( \bigcup_{k < \alpha} I_k J_k \biggl) \subseteq I'_\alpha J'_\alpha \quad \text{where} \ \min(|I'_\alpha|, |J'_\alpha|) \le r \

— The Determinantal Matroid  (2502.18222 - Nicklasson et al., 25 Feb 2025) in Question 3.11, Section 3 (Detecting dependent sets)