Let $n\in\mathbb{N}$ and $X=\{x_1,\dotsb,x_n\}\in\mathbb{R}^n$ be given values. We define symmetrical sums and power sums of $X$ as in Newton identities. So let:
$$s_i(X)=\displaystyle\sum_{\substack{ Y\in X \\ n(Y)=i}}\left(\prod_{y\in Y}y\right)\quad,\quad p_i(x)=\displaystyle\sum_{x\in X}x^i$$
I'm trying to know if $f(X)=\mid x_1\mid+\dotsb+\mid x_n\mid$ can be represented as a linear combination of $s_i$'s and $p_i$'s. Also I'm wondering if there is a way to represent it as a linear combination of $p_i$'s only.
I have reached representations of the expression as a square root expression. As instance:
$$\mid a\mid+\mid b\mid=\sqrt{p_2+2\sqrt{{s_2}^2}}=\sqrt{p_2+\sqrt{({p_1}^2-p_2)^2}}$$
But failed to achieve a polynomial representation of the expression even for $n=2$. On the other hand I believe there should be a representation since the set of infinite $p_i$'s actually determine $X$ uniquely and hence $f(X)$ can be determined having been provided with the set of $p_i$'s.
Any help or reference would be greatly appreciated.