A *symmetric function* is a function $f:\mathbb R^n\to \mathbb R$ such that $f(x\_1,\ldots,x\_n)=f(\sigma(x\_1,\ldots,x\_n))$ for every permutation $\sigma\in S\_n.$


The most commonly encountered symmetric functions are polynomial functions.


**Question 1:** Is it true that, given a symmetric function $f:\mathbb R^n\to \mathbb R,$ there exist symmetric polynomials $p\_1,\ldots,p\_m:\mathbb R^n\to \mathbb R$ and functions $h:\mathbb R^m\to \mathbb R$ and $g\_1,\ldots,g\_n:\mathbb R\to \mathbb R$ such that $f$ can be factorized as $$f(x\_1,\ldots,x\_n)=h(p\_1(g\_1(x\_1),\ldots, g\_1(x\_n)),p\_2(g\_2(x\_1),\ldots, g\_2(x\_n)),\ldots,p\_m(g\_m(x\_1),\ldots, g\_m(x\_n)))?$$


**Question 2:** If $f$ has some regularity, e.g., continuous, $C^\infty$ or analytic, can we find the above $g\_i$ and $h$ to be also regular?


To illustrate the idea behind the question I furnish an example.


If $f:\mathbb R^2\to \mathbb R$ with $$f(x,y)=(x^2+y^2)\ln(\frac{xy}{x+y})+\sin(x)+\sin(y),$$ we have $m=4$, $p\_1(x,y)=x^2+y^2,$ $p\_2(x,y)=xy$, $p\_3(x,y)=x+y$, $p\_4(x,y)=x+y$, $g\_1,g\_2,g\_3=\operatorname{id}\_{\mathbb R}$, $g\_4(x)=\sin(x)$ and $h(x,y,z,t)=x\ln(\frac{y}{t})+t$.


Notice that **Question 1** seems to be true thanks to an argument similar to that in the [answer](https://math.stackexchange.com/a/3800574) of Qi Zhu to [Can every symmetric function be written as some function of a sum?](https://math.stackexchange.com/questions/3800563/can-every-symmetric-function-be-written-as-some-function-of-a-sum).


**Question 3:** does this argument works here?



Can every symmetric function be factorized through symmetric polynomials?
$\newcommand\R{\mathbb R}$For any $(x\_1,\dots,x\_n)\in\R^n$,  

$$P(x):=\prod \_{i=1}^{n}(x-x\_{i})=\sum \_{k=0}^{n}(-1)^{k}e\_k(x\_1,\dots,x\_n)x^{n-k},$$
where the $e\_k$'s are the [elementary symmetric polynomials](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial) in $x\_1,\dots,x\_n$. Therefore and because "the roots of a polynomial are determined by its coefficients", there is a function $g\colon\R^m\to[\R^n]$ such that
$$[x\_1,\dots,x\_n]=g(e\_0(x\_1,\dots,x\_n),\dots,e\_n(x\_1,\dots,x\_n))$$
for all $(x\_1,\dots,x\_n)\in\R^n$, where $m:=n+1$,
$$[x\_1,\dots,x\_n]:=\{(x\_{\pi(1)},\dots,x\_{\pi(n)})\colon\pi\in S\_n\},$$
$$[\R^n]:=\{[x\_1,\dots,x\_n]\colon(x\_1,\dots,x\_n)\in\R^n\},$$
and $S\_n$ is the set of all permutations of the set $\{1,\dots,n\}$.


On the other hand, if a function $f\colon\R^n\to\R$ is symmetric, then there clearly exists a function $u\colon[\R^n]\to\R$ such that
$$f(x\_1,\dots,x\_n)=u([x\_1,\dots,x\_n])$$
for all $(x\_1,\dots,x\_n)\in\R^n$.


Thus,
$$f(x\_1,\dots,x\_n)=h(e\_0(x\_1,\dots,x\_n),\dots,e\_n(x\_1,\dots,x\_n)) \tag{1}\label{1}$$
for all $(x\_1,\dots,x\_n)\in\R^n$, where $h:=u\circ g$. This answers your Question 1. $\quad\Box$




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As an illustration of the previous consideration, here is the particular case of identity \eqref{1} for the symmetric function $f\colon\R^2\to\R$ given by the formula $f(x,y):=x\sin y+y\sin x$:  

$$x\sin y+y\sin x \\
=\Big(-\frac{p(x,y)}2+\sqrt{\frac{p(x,y)^2}4-q(x,y)}\Big) \\
\times\sin\Big(-\frac{p(x,y)}2-\sqrt{\frac{p(x,y)^2}4-q(x,y)}\Big) \\ 
+\Big(-\frac{p(x,y)}2-\sqrt{\frac{p(x,y)^2}4-q(x,y)}\Big) \\
\times\sin\Big(-\frac{p(x,y)}2+\sqrt{\frac{p(x,y)^2}4-q(x,y)}\Big) \\ 
$$
for all $(x,y)\in\R^2$, where $p(x,y):=x+y$ and $q(x,y):=xy$ are elementary symmetric polynomials in $x,y$.




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It is now an easy exercise to show that $h$ is continuous if $f$ is continuous. Hints: (i) note that the polynomial $P$ is monic and use a bound on the roots of a monic polynomial in terms of the bounds on its coefficients; (ii) using compactness and the continuity of the value of a polynomial with respect to the coefficients and the argument of the polynomial, show that the map $g$ is continuous; (iii) the map $f\mapsto u$ is clearly continuous in any reasonable topology; (iv) so, $h=u\circ g$ is continuous.


