Poincare function for moduli of differential-geometric structures

TL;DR

This paper explores the Poincare function's role in counting moduli of local geometric structures, verifies Arnold's conjecture under certain conditions, and introduces new formulas for classification problems in geometry.

Contribution

It establishes the property of the Poincare function related to Arnold's conjecture and derives new counting formulas for differential invariants.

Findings

Verified Arnold's conjecture for algebraic and transitive pseudogroups

Derived new formulas for classification problems in geometry and analysis

Surveyed existing results on differential invariants

Abstract

The Poincare function is a compact form of counting moduli in local geometric problems. We discuss its property in relation to V.Arnold's conjecture, and derive this conjecture in the case when the pseudogroup acts algebraically and transitively on the base. Then we survey the known counting results for differential invariants and derive new formulae for several other classification problems in geometry and analysis.