On the finiteness of differential invariants

Group Foliation of Differential Equations Using Moving Frames

“…The correction terms (32). One of most important features of (33) or (32) is that these equations do not require the coordinate expression of the invariant object to be computed [43].…”

“…That is, the algebra of differential invariants is generated by a finite number of invariants. Modern proofs of the fundamental basis theorem appear in the textbooks [36,47]; other proofs based on Spencer cohomology [23], Weyl algebras [32], homological methods [21], or moving frames [15,39,44] also exist.…”

“…In this section, we introduce the pseudogroup jet differential expressions arising simply from taking the exterior derivative of the pseudogroup jets. Pull-back of these pseudogroup jet differentials by the right moving frame results in an expression for the exterior derivatives of the left moving frame components in terms of 'known quantities': invariant horizontal differential forms and the right moving frame pull-backs of the right Maurer-Cartan forms, computed using the universal recurrence relation (32). Expansion of these exterior derivatives in the invariant horizontal coframe yields differential equations for the left moving frame; after restriction to a resolving system solution, these differential equations become the reconstruction equations.…”

“…The determination of the resolving system relies on the classification of differential invariants and their syzygies, which may be performed algorithmically using the universal recurrence relation (32). The resolving system may be interpreted as a projection of the original differential equation into a space of invariants, accomplished through the application of a right moving frame.…”

Recent Advances in the Theory and Application of Lie Pseudo-Groups

“…For historical contributions to the subject, we refer the reader to the original papers of Lie, Medolaghi, Tresse, and Vessiot, [45,46,58,89,93], for the classical theory of pseudogroups, to Cartan, [16,18], for their reformulation in terms of exterior differential systems, and [25,36,37,43,44,47,61,78,81,86,87,88] for a variety of modern approaches. Recent advances began with [27], that proposed a new approach to the classical theory of moving frames for general transformation groups.…”