Invariant ideals of local analytic and formal vector fields

Abstract

The objective of this paper is to analyse analytic invariant sets of analytic ordinary differential equations (ODEs). For this purpose we introduce semi-invariants and invariant ideals as well as the notion of vector fields in Poincare- Dulac normal form (PDNF). We prove that all invariant ideals of a vector field in PDNF are already invariant for its semi-simple linear part. Additionally, this paper provides a natural characterization of invariant ideals via semi-invariants.