Let us define an equivalence relation $\sim$ on $\mathbb C^n$ by $x \sim y$ if and only if $x = (x_{\sigma(1)}, \dots, x_{\sigma(n)}) = (y_1, \dots,y_n)$ where $\sigma \in \mathbb S^n$ belongs to the permutation group. We define the quotient space $\mathbb C_{sym}^n = \mathbb C^n/\sim$. In Chapter $VI$ of Bhatia's Matrix Analysis, it states: if we define a map $f : \mathbb C_{sym}^n \to \mathbb C^n$ by sending the roots of a monic $n^{th}$ polynomial to its coefficients, then $f$ is a homeomorphism.
Let $g = f^{-1}$. Now suppose I restrict the map $g$ on $\mathbb R^n$. $g |_{\mathbb R^n}: \mathbb R^n \to g(\mathbb R^n)$. Then the image should be a subset in $\mathbb C^n_{sym}$ that is invariant under complex conjugation, i.e., containing complex conjugate pairs. $g|_{\mathbb R^n}$ is certainly a homeomorphism. Is there a place I can find this stated explicitly?