# On Glaeser's Theorem for non-smooth functions

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 $$\mathbb R^n$$?

• Can you provide a reference for Gleaser's theorem? Apr 24, 2020 at 12:46
• For 1), take $F(x_1,\ldots ,x_n)=\sum |x_i|^k$ with $k$ odd.
– abx
Apr 24, 2020 at 13:20
• jstor.org/stable/1970204?origin=crossref&seq=1 Apr 24, 2020 at 15:26
• @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? Apr 24, 2020 at 15:38
• @Robert Israel: maybe I was too hasty. Do you think you could do that for any $k$?
– abx
Apr 24, 2020 at 16:06

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.