How could I find the simplest polynomial $P[x,y]=\sum_{0\leq i,0\leq j} \alpha_{ij} x^i y^j$ with only $N$ local extrema ? I'm mostly interested in the cases $N=2$ and $N=3$.
Meaning there are only $N$ solutions for the system {$\frac{\partial P}{\partial x}=0,\frac{\partial P}{\partial y}=0\}$
For $N=2$ local extrema, a suitable Cassini oval works. For instance, $$ ((x-a)^{2}+y^{2})((x+a)^{2}+y^{2})=b^{4} $$ with $a=1$ and $b^4=0.9$, which gives
For $N=3$, this sextic works: $$ (x^2 +y^2 −1)(x^2 +y^2 −2)(x^2 +y^2 −3)+x^6=0 $$
I don't know whether these are the simplest you can get, but I expect you'll need degree at least $2N$.
I learned about the sextic in the paper Sixty-Four Curves of Degree Six.