Poincaré Function for Moduli of Differential-Geometric Structures

Abstract

The Poincaré 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.

Differential Invariants of Linear Symplectic Actions

“…The Poincaré function P(z) = ∑ ∞ k=0 h k z k is rational in all local problems of analysis according to Arnold's conjecture [15]. In our case, this P(z) differs from m(1 − z) −n by a polynomial reflecting the action of G.…”

Differential invariants of Kundt spacetimes

“…(1−z) d is the Poincaré function, with a polynomial R(z) not divisible by (1 − z), then the asymptotic is encoded by the numbers d and σ = R(1), see [13]. 10 Beware that normalization can result in elements of an algebraic extension of the field of rational invariants.…”

“…(1) 0 on the fiber π −1 1,0 (p 0 ) ∩ E 1 . Here W 2 v is an absolute invariant, with the action transitive on its level sets 13 . As shown in Section 3.2, we can bring any point to the point (omitting equations of p 0 )…”

“…(1−z) d is the Poincaré function, with a polynomial R(z) not divisible by (1 − z), then the asymptotic is encoded by the numbers d and σ = R(1), see [13].…”

Differential invariants of Kundt waves

“…If P (z) = R(z) (1−z) d is the Poincaré function, with a polynomial R(z) not divisible by (1 − z), then the asymptotic is encoded by the numbers d and σ = R(1), see[12] 9. Beware that normalization can result in elements of an algebraic extension of the field of rational invariants.…”