Finiteness properties of formal Lie group actions

TL;DR

This paper proves that matrix coefficients of formal Lie group actions are in a Noetherian ring, extending intersection multiplicity estimates and embedding results for formal diffeomorphisms, thus unifying several previous results in the field.

Contribution

It introduces a new approach showing matrix coefficients belong to a Noetherian ring, extending multiplicity estimates and embedding theorems for formal Lie group actions.

Findings

Matrix coefficients of formal Lie group actions are in a Noetherian ring.

Extended intersection multiplicity estimates to general Lie groups.

Provided a new proof for jet-determination of formal diffeomorphisms.

Abstract

Following ideas of Arnold and Seigal-Yakovenko, we prove that the space of matrix coefficients of a formal Lie group action belongs to a Noetherian ring. Using this result we extend the uniform intersection multiplicity estimates of these authors from the abelian case to general Lie groups. We also demonstrate a simple new proof for a jet-determination result of Baouendi et. al. In the second part of the paper we use similar ideas to prove a result on embedding formal diffeomorphisms in one-parameter groups extending a result of Takens. In particular this implies that the results of Arnold and Seigal-Yakovenko are formal consequence of our result for Lie groups.