I am trying to understand the number of connected components of an algebraic variety $V \subset \mathbb{R}^n$ defined by the system of $m$ polynomial equations and $n>m>1$. The system of equations is defined as follows:
Given $m$ constants $u_k \in \mathbb{R}$, and positive integers $d_k \geq 3 $, $k=1,\cdots,m$,
$ \sum_{i=1}^n x_i^{d_k} = u_k $, $k=1,\cdots,m$.
Note that the equations are invariant under the action of the symmetric group $S_m$. Hence the number of connected component is at most $c \cdot m!$, where $c$ is the number of connected components of the quotient space $V/S_n$.
My question is: how to prove or disprove that the number of connected components of $V/S_n$ is independent of $n$, or Does anybody know some reference about the properties of this class of algebraic varieties.
Thanks in advance.