For $F:\mathbb{R}^2\rightarrow \mathbb{R}$ given by $F(x,y)=x^3+xy+y^3$, how do I show that $F^{-1}(0)$ and $F^{-1}(1/27)$ aren't regular submanifolds? I've plotted these on Wolfram alpha:
the first one crosses itself at a point (so it's not a manifold by the standard "remove this point and see it's got more components than it should" argument)
the second is (edit: the union of) a curve and an isolated point (so it's not a manifold because it doesn't have a well-defined dimension).
But I don't know how to prove these level sets actually look like this. What techniques can I use to work out what they look like?