 0620 Gerard P. Barbanson
 A CHEVALLEY'S THEOREM IN CLASS ${\cal C}^r$.
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Jan 23, 06

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Abstract. Let $W$ be a finite reflection group acting orthogonally on
${\bf R}^n$, $P$ be the Chevalley polynomial mapping determined
by an integrity basis of the algebra of $W$invariant polynomials,
and $h$ be the highest degree of the coordinate polynomials in $P$.
There exists a linear mapping from ${\cal C}^r({\bf R}^n)^W$ to ${\cal C}^{[r/h]}({\bf R}^n)$ such that if $F$ is the image of $f$, $f=F\circ P$. This mapping is continuous for the natural Frechet
topologies. A general counterexample shows that this result is
the best possible. The proof by induction on $h$ uses techniques
of division by linear forms and a study of compensationphenomenons. An extension to $P^{1}({\bf R}^n)$ of invariant formally holomorphic regular fields is needed.
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