Abstract:

Let Γ be a compact Lie group acting on R^n and P_n(Γ) the ring of Γ invariant polynomial germs under Γ. Hilbert-Weyl theorem shows that there is a Hilbert basis consisting of Γ invariant homogeneous polynomial germs for P_n(Γ). However, it is not clear, how to choose a Hilbert basis from Γ invariant homogeneous polynomial germs and how to determine that a finite set of Γ invariant homogeneous polynomial germs is a Hilbert basis of P_n(Γ).  In this paper, by means of some fundamental properties of Noether's ring and invariant integration as well as the relevant theorems in the theory of singularities, a necessary and sufficient condition is  proved for determining a Hilber basis of P_n(Γ). This will provide a new way to determine of a Hilbert basis for some P_n(Γ).

Key words: Compact Lie group, Ring of invariant polynomial germs, Hilbert basis