P_n(Γ)一组Hilbert基的判定和计算

设Γ是一作用在Rn上的紧李群,Pn(Γ)是Γ不变的多项式芽环,Hilbert-Weyl定理证明了对于Pn(Γ)总存在一组由Γ不变的齐次多项式芽构成的Hilbert基.然而,如何从Γ不变的齐次多项式芽中选出一组Hilbert基?如何判定Γ不变的齐次多项式芽的一个有限集就是Pn(Γ)的一组Hilbert基?在有关的文献中,Pn(Γ)的一组Hilbert基常常是通过幂级数展开进行计算.作为一个补充,本文借助Noether环、不变积分的基本性质以及奇点理论的某些定理,证明了判定、计算Pn(Γ)的Hilbert基的有关定理和原理,这提供了计算某些Pn(Γ)一组Hilbert基的而与幂级数展开不同的方法.最后,举例加以说明.

The Criterion and Computation of a Hilbert Basis of Pn(Γ)

Let Γ be a compact Lie group acting on Rn and Pn(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 Pn(Γ). However, questions remain as to 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 Pn(Γ). In the relevant literatures, a Hilbert basis of Pn(Γ) is often computed by using power series expansion. In this paper, by means of the foundamental properties of Noether's ring and invariant integration as well as some theorems in the theory of singularities, the relevant theorems and principles as a complement are proved for determining and computing a Hilbert basis of Pn(Γ). This will provide a method which is different from power series expansion to compute a Hilbert basis of some Pn(Γ). Finally, it will also give some practical examples for explanation.