The conjectures of Kumbhakar, Roy, and Srinivasan

TL;DR

This paper proves conjectures related to the classification of differential equations using differential Galois theory, demonstrating their validity over constant fields and limitations over nonconstant fields, and connecting recent results in the field.

Contribution

It establishes the validity of conjectures on differential equations classification via Galois theory and links recent advances in abelian reductions to these conjectures.

Findings

Conjectures are proven over constant fields.

Stronger versions hold over the field of constants.

Results cannot be improved over nonconstant differential fields.

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.