
> 
> By Bézout's theorem a quartic polynomial $p(x,y)$ can have at most 9 isolated critical points. Can all of them be saddle points?
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In case of a cubic polynomial there is a mechanical way to answer this type of questions: One can find a general form of such polynomials with critical points at three given points, e.g. at $(0,0),(0,1),$ and $(1,0)$, and then play with the remaining free parameters until a polynomial with the desired properties if found, as there is only one remaining critical point whose position was not normalized by affine transformation [[1]](https://mathoverflow.net/q/446848/497175), [[2]](https://math.stackexchange.com/q/4701058/1134951).


However, the above approach won't work in case of a quartic polynomial, as then affine transformation normalizes position of only 3 out of the 9 potential critical points.


Note that a quartic polynomial in two real variables can have at most 5 minima out of its 9 potential critical points [[3]](https://math.stackexchange.com/q/4620663/1134951), [[4]](https://mathoverflow.net/q/442736/497175). Can it reversely have no extreme points, so that all 9 of the potential critical points would be saddle points?



How many saddle points can a quartic polynomial in two real variables have? All 9?
By (3.1) of [Counting Critical Points of Real Polynomials in Two Variables by Alan Durfee, Nathan Kronefeld, Heidi Munson, Jeff Roy, Ina Westby](https://www.jstor.org/stable/2324459?seq=7) a degree $d$ polynomial with only nondegenerate critical points can contain at most $d(d-1)/2$ saddle points. For $d=4$ this gives a max of $6$ saddle points.


This bound is attained by a product of $d$ linear polynomials, as long as each pair of lines intersects at a different point, as then the $d(d-1)/2$ intersection points are all saddle points.


(This paper was cited in a paper cited in [one of the answers you cite](https://math.stackexchange.com/questions/4620663/how-many-strict-local-minima-a-quartic-polynomial-in-two-variables-might-have).)


