Can a cubic polynomial in two real variables have exactly three isolated critical points?

Question

Is there a cubic polynomial $c(x,y)$ with exactly 3 critical points?

A cubic polynomial can have:

• 0 critical points, e.g. $x^3+y$;
• 1 critical point, e.g. $x^3+y^3$;
• 2 critical points, e.g. $x^3 + y^3-y$;
• 4 critical points, e.g. $x^3-x + y^3-y$.

Can the number be 3?

Observations:

• By Bézout's Theorem a cubic polynomial (in two variables) has at most 4 isolated critical points.
• It is easy to see that a cubic polynomial has at most one maximum / minimum. Thus every polynomial $c(x,y)$ with 4 critical points has at least 2 saddle points. For example, the polynomial $c(x,y) = x^3-3x + y^3-3y$ has four critical points: Local maximum at $(-1,-1)$,

Closely related question whether a cubic polynomial in two variables can have 3 saddle points: https://mathoverflow.net/q/446848/497175

Answer 1

Yes. The cubic polynomial $$ c(x,y) := 3x^3 + x^2y + y^3 + x^2 + 2xy + y^2 $$ has 3 isolated critical points at $(0,0)$, $(-\frac 6 {13}, - \frac{12}{13})$ and $(\frac 2 {19}, - \frac{8}{57})$.

The gradient is given by $$ \begin{align} c_x(x,y) & = 9x^2 + 2xy + 2x + 2y \\ c_y(x,y) & = x^2 + 3 y^2 + 2x + 2 y \end{align} $$

My idea to find this example was to see the set of critical points at the intersection of two conics given by $c_x = 0$ and $c_y = 0$. If you want two conics to have 3 intersection points, one of them needs to be a tangent contact point. Hence, at this point, the gradients of $c_x$ and $c_y$ must be colinear. By translation, I assumed this double point to be $(0,0)$. This allowed by to set the coefficients of order $0$, $1$ and $2$ of $c$. I then chose the leading coefficients by trial and error.

For the example given above, $c_y = 0$ is an ellipse and $c_x = 0$ is an hyperbola.

It is hard to tell on the graph of the two conics, but there are indeed two intersection points in the region where they overlap. You can visualize this better by expressing the curves $y_{red}(x)$ such that $c_y(x,y_{red}(x)) = 0$ and $y_{blue}(x)$ similarly and plotting $y_{red}-y_{blue}$.