Plane Polynomials and Hamiltonian Vector Fields Determined by Their Singular Points

Abstract

Let $$\Sigma (f)$$ Σ ( f ) be the singular points of a polynomial $$f \in \mathbb {K}[x,y]$$ f ∈ K [ x , y ] in the plane $$\mathbb {K}^2$$ K 2 , where $$\mathbb {K}$$ K is $$\mathbb {R}$$ R or $$\mathbb {C}$$ C . Our goal is to study the singular point map $$\mathfrak {S}_d$$ S d , it sends polynomials f of degree d to their singular points $$\Sigma (f)$$ Σ ( f ) . Very roughly speaking, a polynomial f is essentially determined when any other g sharing the singular points of f satisfies that $$f = \lambda g$$ f = λ g ; here both are polynomials of degree d, $$\lambda \in \mathbb {K}^* $$ λ ∈ K ∗ . In order to describe the degree d essentially determined polynomials, a computation of the required number of isolated singular points $$\delta (d)$$ δ ( d ) is provided. A dichotomy appears for the values of $$\delta (d)$$ δ ( d ) ; depending on a certain parity, the space of essentially determined polynomials is an open or closed Zariski set. We compute the map $$\mathfrak {S}_{3}$$ 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 $$\hbox { Aff}\hspace{1.42271pt}(\mathbb {K}^2)$$ Aff ( K 2 ) determines a singular $$\mathbb {K}$$ K -analytic surface.