About finite differential tropical basis for linear ODE's

Summary
This paper investigates the tropicalization of linear ordinary differential equations to calculate the radius of convergence of classical solutions. It explores conditions for the existence of a finite differential tropical basis, focusing on second- and third-order cases with distinct root valuations.

Mathematical/empirical assessment
The derivations connecting tropical vanishing conditions to convergence radii are sound. The step-by-step resolution of the recurrence relations in Eq. (7) and the stratification in Fig. (2) provide a clear combinatorial picture of the solution spaces.

Strengths
The transition from trivial to non-trivial valuations is handled elegantly. Building on "Initial forms and a notion of basis for tropical differential equations", the work successfully identifies sufficient conditions for finite bases. The geometric intuition provided by the Bergman fan in Fig. (1) is a wonderful pedagogical tool that really helps ground the abstract concepts.

Concerns
While the extension to arbitrary order is mentioned as straightforward, the notation becomes quite cumbersome. From my own experience writing in this area, developing a more abstract theoretical framework for the general order r case would greatly strengthen the narrative and save the reader from getting lost in the indices. Additionally, the behavior when roots share the same valuation remains open, which is a natural limitation for this stage of the project.

Final decision
Weak accept