TITLE: Is the set of points in $\mathbb R^n$ with $\sum_{j=1}^n x_j^k = 0$ a submanifold?
QUESTION [7 upvotes]: Consider the set
$$A:= \{x\in \mathbb R^n :\sum_{j=1}^n x_j^k = 0\}$$
for $k$ an odd integer. Is this a submanifold of $\mathbb R^n$ for every $n$? For $n=1$, it is just 0; for $n = 2$, it is the anti-diagonal $\{(x_1 , -x_1) : x_1 \in \mathbb R\}$, which is a submanifold. However, I cannot find a way to determine this in higher dimensions. Any suggestions?

REPLY [4 votes]: Your set is scale-invariant: if $x\in A$ and $t\in\mathbb R$ then $tx\in A$. If such a set is a submanifold, then 
it is a linear subspace. If $n>2$, $A$ is not a linear subspace, hence not a manifold.