Gröbner Bases, Invariant Theory and Equivariant Dynamics

This paper is about algorithmic invariant theory as it is required within equivariant dynamical systems. The question of generic bifurcation equations (arbitrary equivariant polynomial vector) requires the knowledge of fundamental invariants and equivariants. We discuss computations which are related to this for finite groups and semi-simple Lie groups. We consider questions such as the completeness of invariants and equivariants. Efficient computations are gained by the Hilbert series driven Buchberger algorithm because computation of elimination ideals is heavily required. Applications such as orbit space reduction are presented.