Plane Polynomials and Hamiltonian Vector Fields Determined by Their Singular Points

Let Σ (f) Σ ( f ) be the singular points of a polynomial f ∈ K[x,y] f ∈ K [ x , y ] in the plane K² K 2 , where K K is R R or C C . Our goal is to study the singular point map Sd S d , it sends polynomials f of degree d to their singular points Σ (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 = λ g f = λ g ; here both are polynomials of degree d , λ ∈ K^* λ ∈ K ∗ . In order to describe the degree d essentially determined polynomials, a computation of the required number of isolated singular points δ (d) δ ( d ) is provided. A dichotomy appears for the values of δ (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 S₃ 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 Aff1.42271pt(K²) Aff ( K 2 ) determines a singular K K -analytic surface.