06-244 Gerard P. BARBANSON

A CHEVALLEY'S THEOREM IN CLASS ${\cal C}^r$. (253K, pdf) Sep 3, 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: ${\cal C}^r(R^n)^W \ni f \to F\in {\cal C}^{[r/h]}(R^n)$ such that $f=F\circ P$, continuous for the natural Fr\'echet 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 compensation phenomenons. An extension to $P^{-1}({\bf R}^n)$ of invariant formally holomorphic regular fields is needed. This is a revised version of # 06-20 posted earlier on this site with the same name.

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