Involutive moving frames II; The Lie-Tresse theorem

TL;DR

This paper advances the theory of involutive moving frames by providing a new proof of the Lie-Tresse theorem, establishing bounds on differential invariants, and demonstrating computational benefits for PDE equivalence problems.

Contribution

It introduces a unified framework for involutive moving frames, offers a new proof of the Lie-Tresse theorem, and applies this to PDE equivalence problems.

Findings

New constructive proof of the Lie-Tresse theorem

First general upper bound on minimal differential invariants

Demonstrated computational advantages in PDE equivalence

Abstract

This paper continues the project, begun in \cite{IMF}, of harmonizing Cartan's classical equivalence method and the modern equivariant moving frame in a framework dubbed \emph{involutive moving frames}. As an attestation of the fruitfulness of our framework, we obtain a new, constructive and intuitive proof of the Lie-Tresse theorem (Fundamental basis theorem) and a first general upper bound on the minimal number of generating differential invariants for Lie pseudo-groups. Further, we demonstrate the computational advantages of this framework by studying the equivalence problem for first order PDE in two independent variables and one dependent variable under point transformations.