13A50-Schwarz1975Theorem.tex

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\pmtitle{Schwarz (1975) theorem}
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\textbf{theorem:}
\begin{quote}
Let $\Gamma$ be a compact Lie group acting on $V$.  Let $u_1,\ldots ,u_s$ be a Hilbert basis for the $\Gamma$-invariant polynomials $\mathcal{P}(\Gamma )$ \textit{(see Hilbert-Weyl theorem)}.  Let $f\in \mathcal{E}(\Gamma )$.  Then there exists a smooth germ $h\in \mathcal{E}_s$ \textit{(the ring of $C^{\infty}$ germs $\mathbb{R}^s \to \mathbb{R}$)} such that $f(x) = h(u_1(x),\ldots , u_s(x))$. \cite{1}
\end{quote}
\textbf{proof:}
\begin{quote}
The proof is shown on page 58 of \cite{1}.
\end{quote}
\textbf{theorem:} \textit{(as stated by Gerald W. Schwarz)}
\begin{quote}
Let $G$ be a compact Lie group acting orthogonally on $\mathbb{R}^n$, let $\rho_1 ,\ldots, \rho_k$ be generators of $\mathcal{P}(\mathbb{R}^n)^G$ \textit{(the set $G$-invariant polynomials on $\mathbb{R}^n$)}, and let $\rho = (\rho_1 ,\ldots, \rho_k):\mathbb{R}^n \to \mathbb{R}^k$.  Then $\rho*\mathcal{E}(\mathbb{R}^k) = \mathcal{E}(\mathbb{R}^n)^G$. \cite{2}
\end{quote}
\textbf{proof:}
\begin{quote}
The proof is shown in the following publication \cite{2}.
\end{quote}
\begin{thebibliography}{2}
\bibitem[GSS]{1} Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David: Singularities and Groups in Bifurcation Theory \textit{(Volume II)}. Springer-Verlag, New York, 1988.
\bibitem[SG]{2} Schwarz, W. Gerald: Smooth Functions Invariant Under the Action of a Compact Lie Group, \textit{Topology} Vol. 14, pp. 63-68, 1975.
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