What would be a systematic way to find a cubic polynomial $p(x,y)$ with 4 saddle points?
A cubic polynomial can have at most 4 critical points. Note that no more than 2 critical points can lie on a line (since $c$ along a line is a cubic polynomial in one variable and thus can have at most two critical points). Therefore, any cubic polynomial with 4 critical point can be modified by an affine transformation so that it has critical points $(0,0),(0,1),(1,0)$. Normalizing $c(0,0)=0$ in addition to it, we get the following formula for all polynomials with the mentioned properties: $$ p(x, y) = a\left(x^3 - \frac{3}{2}x^2\right) + b\left(y^3 - \frac{3}{2}y^2\right) + c\left(x^2y + xy^2 - xy\right). $$ What remains is to choose $a,b,c\in \mathbb R$ so that:
- The determinant of the Hessian at the three mentioned points is negative.
- The polynomial has a 4th critical point $(u,v)$.
- The determinant of the Hessian is negative at $(u,v)$.
At this point one could use some "frute-force" approach, like to randomly keep generating $a,b,c$ until all three conditions were satisfied, however I would be interested in a more systematic method:
How to find an example of $(a,b,c)\in \Bbb R^3$ satisfying the nonlinear inequalities implied by conditions 1.-3. using a freely available program?
Related:
