判定P_n(Γ)的Hilbert基的一个充要条件

A Necessary and Sufficient Condition for Determining a Hilbert Basis of P_n(Γ)

设Γ是一作用在RR上的紧李群,Pn(Γ)是Г不变的多项式芽构成的环.Hilhert-Weyl定理证明了对于Pn(Γ)总存在一组由Г不变的齐次多项式芽组成的Hilbert基.然而,如何从Г不变的齐次多项式芽中选出一组Hilbert基?如何判定Г不变的齐次多项式芽的一个有限集就是Pn(Γ)的一组Hilbert基?该文借助于Noether环和不变积分的某些基本性质以及奇点理论的有关定理,证明了判定Pn(Γ)的Hilbett基的一个充要条件.这对某些Pn(Γ)提供了计算一组Hilbert基的新途径.

Let Г be a compact Lie group acting on Rn and Pn(Г) the ring of Г invariant polynomial germs under Г. Hilbert-Weyl theorem shows that there is a Hilbert basis consisting of Г invariant homogeneous polynomial germs for Pn(Г) . However, it is not clear,how to choose a Hilbert basis from Г invariant homogeneous polynomial germs and how to determine that a finite set of Г invariant homogeneous polynomial germs is a Hilbert basis of Pn(Г) . 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 Hilbert basis of Pn(Г) . This will provide a new way to determine of a Hilbert basis for some Pn(Г) .