Let \(\Sigma (f)\) be the singular 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 singular point map \(\mathfrak {S}_d\) , it sends polynomials f of degree d to their singular points \(\Sigma (f)\) . Very roughly speaking, a polynomial f is essentially determined when any other g sharing the singular points of f satisfies that \(f = \lambda g\) ; here both are polynomials of degree d, \(\lambda \in \mathbb {K}^* \) . In order to describe the degree d essentially determined polynomials, a computation of the required number of isolated singular 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 4 points leads to a degree 3 essentially determined polynomial. Furthermore, we describe explicitly configurations supporting degree 3 non essential determined polynomials. The quotient space of essentially determined polynomials of degree 3 up to the action of the affine group \(\hbox { Aff}\hspace{1.42271pt}(\mathbb {K}^2)\) determines a singular \(\mathbb {K}\) -analytic surface.
