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What number of critical points all of which are saddle points can a cubic polynomial $c(x,y)$ in two real variables have?

Note: We restrict attention only to non-degenerate critical points.

It follows from the Bézout's theorem that a cubic polynomial can have at most 4 critical points. It is easy to see that it can have 0,1,2, or 4 critical points and it can have 3 of them is questioned in: Can a cubic polynomial in two real variables have exactly three isolated critical points?

Observations:

  • A polynomial with no critical points has no saddle points, e.g. $x^3+y$.
  • If we did not insist on the polynomial having the cubit term, then $x^2 - y^2$ could serve as an example of a polynomial with a single saddle point, however adding a cubic term usually introduces other critical points.
  • An example of a polynomial with two critical points out of which both are saddle points is $c(x,y)= x^{(2)} (y+1)-y^{(2)} (x+1)$, with saddles at $(0,0)$ and $(-2,-2)$.

A closely related question whether a cubic polynomial in two variables can have 3 saddle points: https://mathoverflow.net/q/446848/497175. Whether a cubic polynomial can have three critical points is asked here: Can a cubic polynomial in two real variables have exactly three isolated critical points?

Paul Sinclair
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Pavel Kocourek
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    The cubic $$f(x,y)=2 x^3 + 3 x^2 + 12 y^2 - 12 x - 6 x y^2$$ has four critical points at $(-2,0)$, $(1,0)$, $(2,-2)$, $(2,2)$. All but the second are saddles. A couple years ago I spent quite a bit of time cooking up examples like this, where all the critical points have integer coordinates, and they can also be found with very basic techniques like here. I needed scores of such examples when I switched to giving my students individual problems instead of final exams (that I no longer could justify without compromising intellectual honesty, COVID also played a role). – Jyrki Lahtonen May 16 '23 at 18:43

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