Poénaru (1976) theorem

Let $\mathrm{\Gamma }$ be a compact Lie group and let $g_1,\mathrm{\dots },g_r$ generate the module $\overrightarrow{𝒫}(\mathrm{\Gamma })$(the space of $\mathrm{\Gamma }$-equivariant polynomial mappings) of $\mathrm{\Gamma }$-equivariant polynomials over the ring $𝒫(\mathrm{\Gamma })$(the ring of $\mathrm{\Gamma }$-invariant polynomial). Then $g_1,\mathrm{\dots },g_r$ generate the module $\overrightarrow{ℰ}(\mathrm{\Gamma })$(the space of $\mathrm{\Gamma }$-equivariant germs at the origin of $C^{\mathrm{\infty }}$ mappings) over the ring $ℰ(\mathrm{\Gamma })$(the ring of $\mathrm{\Gamma }$-invariant germs). [GSS]