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Glaeser's Theorem says that a $C^\infty$ function $F$ on $\mathbb R^n$ which is invariant under permutation of the variables is a smooth function of the symmetric polynomials of $(x_1, \dots, x_n)$.

Question 1: What remains (if anything) of this statement if $F$ is $C^k$ ?

Question 2: In the statement above, is it clear that you can replace $\mathbb R^n$ by a symmetric open subset of $\mathbb R^n$?

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    $\begingroup$ Can you provide a reference for Gleaser's theorem? $\endgroup$ Apr 24 '20 at 12:46
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    $\begingroup$ For 1), take $F(x_1,\ldots ,x_n)=\sum |x_i|^k$ with $k$ odd. $\endgroup$
    – abx
    Apr 24 '20 at 13:20
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    $\begingroup$ jstor.org/stable/1970204?origin=crossref&seq=1 $\endgroup$ Apr 24 '20 at 15:26
  • $\begingroup$ @abx Your example is $C^{k-1}$ and invariant under permutation. Is it not a $C^{k-1}$ function of the symmetric polynomials? In the case $n=2$, $k=1$, $|x_1| + |x_2| = \sqrt{(x_1+x_2)^2 - 2 x_1 x_2 + 2 |x_1 x_2|}$. Or are you just saying it's not a smooth function? $\endgroup$ Apr 24 '20 at 15:38
  • $\begingroup$ @Robert Israel: maybe I was too hasty. Do you think you could do that for any $k$? $\endgroup$
    – abx
    Apr 24 '20 at 16:06
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A more direct reference (quoted in Rumberger's paper) is: G. Barbançon, Le théorème de Newton pour les fonctions de classe $C^r$. Ann. Sci. École Norm. Sup. 5 (1972), 435–458. He proves that a symmetric function of class $C^{nr}$ on $\Bbb{R}^n$ is a function of class $C^{r}$ of the symmetric polynomials.

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My collegue Armin Rainer pointed me to the following paper, which has some answers, also in its references. There is a loss of differentiability involved.

  • Matthias Rumberger: Finitely differentiable invariants. Math. Z. 229, 675–694 (1998)
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