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