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The conjectures of Kumbhakar, Roy, and Srinivasan

James Freitag, Omar León Sánchez, Wei Li, Joel Nagloo

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.

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.
Paper Structure (11 sections, 25 theorems, 6 equations)

This paper contains 11 sections, 25 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. A note on authorship
  3. Notions from model theory
  4. Order one differential equations
  5. Improvements to the conjecture
  6. On the optimality of the conjecture of Kumbhakar, Roy, and Srinivasan
  7. Higher order equations - the Kumbhakar-Srinivasan conjecture
  8. Some remarks on the proof of Theorem \ref{['thegeneral']}.
  9. On the optimality of the conjecture of Kumbhakar and Srinivasan
  10. Abelian reduction and the structure of $k$-irreducible subfields
  11. The inverse problem

Key Result

Theorem 1.3

Let $k$ be an algebraically closed differential field and $y$ a nonalgebraic solution of (arbintro).

Theorems & Definitions (46)

  • Conjecture 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Definition 3.1
  • Lemma 3.2
  • ...and 36 more