  Let $\Sigma(f)$ be critical points of a polynomial $f \in \mathbb{K}[x,y]$ in
the plane $\mathbb{K}^2$, where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$.
Our goal is to study the critical point map $\mathfrak{S}_d$, by sending
polynomials $f$ of degree $d$ to their critical points $\Sigma(f)$ . Very
roughly speaking, a polynomial $f$ is essentially determined when any other $g$
sharing the critical points of $f$ satisfies that $f= \lambda g$; here both are
polynomials of at most degree $d$, $\lambda \in \mathbb{K}^*$. In order to
describe the degree $d$ essentially determined polynomials, a computation of
the required number of isolated critical points $\delta (d)$ is provided. A
dichotomy appears for the values of $\delta (d)$; depending on a certain parity
the space of essentially determined polynomials is an open or closed Zariski
set. We compute the map $\mathfrak{S}_3$, describing under what conditions a
configuration of four points leads to a degree three essentially determined
polynomial. Furthermore, we describe explicitly configurations supporting
degree three non essential determined polynomials. The quotient space of
essentially determined polynomials of degree three up to the action of the
affine group $\hbox{Aff}(\mathbb{K}^2)$ determines a singular surface over
$\mathbb{K}$.
