# Local invariant sets of analytic vector fields

- Date:
- 27.05.19
- Times:
- 15:30
- Place:
- UAB - Dept. Matemàtiques (C1/-128)
- Speaker:
- Niclas Kruff
- University:
- Aachen University

**Abstract**

In the theory of autonomous ordinary differential equations invariant sets play an important role. In particular, we are interested in local analytic invariant sets near stationary points. Invariant sets of a differential equation correspond to invariant ideals of the associated derivation in the power series algebra. Poincaré-Dulac normal forms are very useful in studying semi-invariants and invariant ideals. We prove that an invariant ideal with respect to a vector field, given in normal form, is already invariant with respect to the semi-simple part of its Jacobian at the stationary point. This generalizes a known result about semi-invariants, that is invariantsets of codimension 1. Moreover, we give a characterization of all ideals which areinvariant with respect to the semi-simple part of the Jacobian. As an application, we consider polynomial systems and we provide a sharp bound of the total degree of possible polynomial semi-invariants under some generic conditions