Determine the sharp order of the consecutive dependence count

Determine whether the upper bound M(H) \ll H^{1/2}(\log H)^c for the number M(H) of consecutive multiplicatively dependent triples of maximal rank is tight, and, if it is not, improve the bound, potentially by analyzing quadratic factors of the resultants F_{\boldsymbol{t}}(X,Y) and G_{\boldsymbol{t}}(X,Y).

Background

The paper proves an upper bound of order H{1/2} up to logarithmic factors for consecutive multiplicatively dependent triples of maximal rank. The authors state that they do not believe this bound is optimal and suggest that studying quadratic factors of the resultants arising in their argument could lead to an improvement. They also note that this direction appears substantially more difficult than their analysis of linear factors.

References

We do not believe that the bound of Theorem~\ref{thm:Bound MH} is tight and perhaps investigating possible quadratic factors of the resultants $F_\vt(X,Y)$ and $G_\vt(X,Y)$ is one of the ways to improve our bound.

— Counting consecutive multiplicatively dependent triples  (2609.29408 - Shparlinski et al., 24 Sep 2026) in Section 4, Open questions

In fact, it is not even clear whether $M(H) \to \infty$ as $H\to \infty$, seeQuestion~4.

— Counting consecutive multiplicatively dependent triples  (2609.29408 - Shparlinski et al., 24 Sep 2026) in Section 4, Open questions