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 semiinvariants and invariant ideals as well as the notion of vector fields in Poincaré-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.

Symmetries of Ordinary Differential Equations: A Short Introduction

“…The following are two main results from the doctoral dissertation [22] of N. Kruff. The proofs are too long (and a bit too technically involved) to be presented here.…”

Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

“…These properties hold more generally for polynomials. An algebraic proof of this fact (which is known) is given in [19], Lemma 8.4, and we give an elementary ad-hoc proof in Sect. 1 of the Appendix.…”

“…Let Cv correspond to one of these points. Then, by Theorems 3.1 and 3.2 of [20], the local ideal defining the image of the curve under the Poincaré transform is generated by certain semi-invariants of D f * v (0), and there must be n − 1 of these, since the dimension of the curve equals one. Using property E and Lemma 3, and the fact that the transformed curve is not contained in the hyperplane {x : x n+1 = 0}, one sees that the only possible ideal is the one not containing the semi-invariant x n+1 .…”

“…The purpose of the present paper, which is based in part on the doctoral thesis [19] by one of the authors, is to generalize the results of [32] to higher dimensions, with a focus on dimension three. The emphasis is on establishing verifiable conditions that provide uniform degree bounds for irreducible semi-invariants.…”

Attracting and Natural Invariant Varieties for Polynomial Vector Fields and Control Systems

“…Recently, some progress has been made in the area of nonlinear polynomial systems; see [15][16][17]21], and Yuno and Ohtsuka [19,20], where it is shown that methods from symbolic computation can be used to test the conditions for controlled invariance of varieties constructively. The purpose of the present work, which is in part based on the doctoral dissertations [11,17] by two authors of the present manuscript, is twofold. First, for real systems (1) there is interest not only in invariance of algebraic varieties but also in attractiveness.…”