Scalar relative differential invariants

TL;DR

This paper studies the algebraic structure of scalar relative differential invariants, showing finite generation after localization and analyzing their weights and orders with various examples.

Contribution

It demonstrates that the algebra of polynomial differential invariants is not finitely generated, but becomes finitely generated after localization, and provides bounds on invariant weights and order.

Findings

Algebra of polynomial differential invariants is not finitely generated.

Localization on a finite set yields finite generation.

Bounds established for weights and order of invariants.

Abstract

Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their (differential) algebra and demonstrate both positive and negative results in this respect under various setups. As in the algebraic case, the algebra of polynomial differential invariants is not finitely generated. However we show that after localization on a finite set of relative invariants the differential algebra becomes finitely generated. We also investigate the weights of rational relative differential invariants and bound their order. Several nontrivial examples are considered and further applications are discussed.