Finding singular points of a non-algebraic curve.

[jdinatale]

Let $F : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the map given by $F(x, y) := (x^3 - xy, y^3 - xy)$. What are some singular points?

Well, I know that for an algebraic curve, a point $p_0 = (x_0, y_0)$ is a singular point if $F_x(x_0, y_0) = 0$ and $F_y(x_0, y_0) = 0$.

However, this curve is not algebraic, so I'm not sure if that still applies. If it does, then

$F_x(x, y) = (3x^2 - y, -y) = (0, 0)$ and $F_y(x, y) = (-x, 3y^2 - x) = (0, 0)$ at the point $p_0 = (0, 0)$

Is that the correct way of determining the singular points? Are there any others?

I graphed it in Mathematica.

 

[micromass]

You don't really have a curve here, but rather something like an algebraic surface. Does your text say how singular points are defined in a surface?? There should probably be a determinant condition.