The conjectures of Kumbhakar, Roy, and Srinivasan

Abstract

We prove a conjecture of Kumbhakar, Roy, and Srinivasan (2024) on the classification of order one differential equations, and a conjecture of Kumbhakar and Srinivasan (2025) on higher order equations. Both conjectures are shown to be results of recent work in differential Galois theory. In both cases, stronger versions of the conjectures hold when working over the field of constants. We use inverse Galois theory to show the conjectures cannot be improved over any nonconstant differential field. We also show how certain recent results of Jaoui and Moosa (2024) on abelian reductions of differential equations can be recovered from the work of Kumbhakar and Srinivasan (2025) and vice versa.