Symmetric functions and regularity (II) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:01:41Z http://mathoverflow.net/feeds/question/61795 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61795/symmetric-functions-and-regularity-ii Symmetric functions and regularity (II) Denis Serre 2011-04-15T09:25:10Z 2011-04-15T15:51:58Z <p>My previous question (where <a href="http://mathoverflow.net/questions/61720" rel="nofollow">$n=2$</a>) was a bit too naive. I think that this one, which is the one being of genuine interest to me, is more involved.</p> <p>Let $f=\mathbb R^n\rightarrow\mathbb R$ be a symmetric function: $$f(x^\sigma)=f(x),\qquad\forall\sigma\in\frak S_n,$$ where $x^\sigma:=(x_{\sigma(1)},\ldots,x_{\sigma(n)})$. There exists a unique function $F$, defined on the image of $\mathbb R^n$ under the map $x\mapsto (p_1,\ldots,p_n)$, where $p_j$ is the elementary symmetric polynomial of degree $j$, such that $$f(x)=F(p_1,\ldots,p_n).$$</p> <blockquote> <p>What is the regularity of $F$ ? Does $f\in\mathcal C^r$ imply $F\in\mathcal C^{r/n}$ and not more ?</p> </blockquote> <p><em>Edit</em>. By $\mathcal C^s$-regularity on a closed domain, one means as usual that the function under consideration is the restriction of a $\mathcal C^s$-function defined on an open super-set.</p>