When is a curve parametrizable? Is there a way in general to tell whether a given curve is parametrizable?


Suppose you are considering the level set $$f(x,y)=c$$ and we want to study whether we can parametrize this curve near the point $(x_0,y_0)$, which is a point in it, $f(x_0,y_0)=c$. If one of the partial derivatives is non-zero at that point, say $\frac{\partial f}{\partial y}(x_0,y_0)\neq0$ then we can use the Implicit function theorem to argue that there is such a local parametrization.
The implicit function theorem can be used for any number of variables.

Example: Suppose we study $$\sin(y)+y-xe^x=0$$ and we want to see if there is local parametrization at the origin $(0,0)$. We see that $$\frac{\partial}{\partial y}(\sin(y)+y-xe^x)|_{(0,0)}=(\cos(y)+1)|_{(0,0)}=2\neq0$$ Therefore, by the Implicit function theorem there is a function $g(x)$ such that $$\sin(g(x))+g(x)-xe^x=0$$ for all $x$ in a neighborhood of $x=0$.

When $f(x,y)$ is a polynomial sometimes the implicit function theorem is not applicable because the condition on the derivatives are not met.

Example: If we look at $x^2-y^3=0$ at the origin $(0,0)$ both partial derivatives are zero. Still, a parametrization exists $t\mapsto(t^3,t^2)$.

It is a very hard-to-prove theorem that in the case of polynomials nice parametrizations like this always exist.

