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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