Point Equivalence of Second-Order ODEs: Maximal Invariant Classification Order

TL;DR

This paper characterizes the local equivalence of second-order ODEs using differential invariants up to order 10, identifying the Painlevé-I equation as a key example requiring maximal jet order for classification.

Contribution

It establishes the exact order bound for differential invariants needed for classifying second-order ODEs under point transformations and links Painlevé-I to this maximal invariant complexity.

Findings

Differential invariants of order at most 10 suffice for classification.

Painlevé-I is uniquely characterized among second-order ODEs by requiring 10th order jets.

The bound of 10 is proven to be sharp, indicating optimality.

Abstract

We show that the local equivalence problem for second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also show that, modulo Cartan duality and point transformations, the Painlev\'e-I equation can be characterized as the simplest second-order ODE belonging to the class of equations requiring 10th order jets for their classification.