Consider the following system $$x'=y-p(x)\\y'=x-q(y)$$ Where $p$ and $q$ are two arbitrary polynomials of order $n$. What is the maximum number of critical points the system can have? Argue why this number of critical points is reached.
My attempt:
$P(x,y)=y-p(x)$ is a polynomial of order $n$ and therefore it has at maximum $n$ roots. The same happens for $Q(x,y)=y-q(x)$, therefore we can have at maximum $n$ critical points.
I don't know how to argue when the system reaches those $n$ critical points.
The solution points have by the first equation the form $(x,p(x))$. Inserting this into the second equation they have to satisfy $$0=x-q(y)=x-q(p(x)).$$ The composite polynomial has degree $\deg p\cdot\deg q$, thus the resulting polynomial equation has that many roots. Only the real roots result in critical points for the real-valued dynamical system.