Elie Cartan's geometrical vision or how to avoid expression swell

TL;DR

The paper tackles the difficulty of solving equivalence problems via direct differential elimination, arguing for Cartan's method as a superior approach. Using their Cartan-based implementation, the authors establish two new equivalence results: a system of $2^{nd}$ order ODEs is equivalent to a flat system (i.e., $\'\ddot{x}=0$), and a system of holomorphic PDEs with two independent variables and one dependent variable is flat. They show that finding a transformation to a target equation becomes algebraic when the target's symmetry pseudogroup is zero-dimensional, and they avoid expression swell by employing non-commutative derivations tailored to the problem. The work demonstrates the practical advantages of Cartan's method over direct methods and provides constructive criteria and techniques for deriving equivalence and flatness results within a non-swell computational framework.

Abstract

The aim of the paper is to demonstrate the superiority of Cartan's method over direct methods based on differential elimination for handling otherwise intractable equivalence problems. In this sens, using our implementation of Cartan's method, we establish two new equivalence results. Weestablish when a system of second order ODE's is equivalent to flat system (second derivations are zero), and when a system of holomorphic PDE's with two independent variables and one dependent variables is flat. We consider the problem of finding transformation that brings a given equation to the target one. We shall see that this problem becomes algebraic when the symmetry pseudogroup of the target equation is zerodimensional. We avoid the swelling of the expressions, by using non-commutative derivations adapted to the problem.