Let $f(x,y)=x^3+xy+y^3+1$. Show that $f^{-1}(f(0,0))$ is not an embedded submanifold.

Question

I have shown that $f^{-1}(f(1/3,1/3))$ and $f^{-1}(f(-1/3,-1/3))$ are embedded submanifolds. In the first case, I calculated the set $\{f^{-1}(f(1/3,1/3))\}$ directly and in the second case, I showed that the Jacobian was onto. However, I am not sure how to prove that $f^{-1}(f(0,0))$ is not an embedded submanifold since the Jacobian is zero at this point and I am having trouble explicitly calculating the set $\{f^{-1}(f(0,0))\}$. Is there some other way to show that this is not an embedded submanifold?

Answer 1

To visualize what this looks like, recognize that the level set $f^{-1}(1)$ is the so-called "folium of Descartes". At the origin, the curve is self-intersecting, which is intuitively why it is not an embedded submanifold. There will be no neighborhood of the origin in which it is locally Euclidean because of this intersection.