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- Title
Plane Polynomials and Hamiltonian Vector Fields Determined by Their Singular Points.
- Authors
Arredondo, John A.; Muciño-Raymundo, Jesús
- Abstract
Let Σ (f) be the singular points of a polynomial f ∈ K [ x , y ] in the plane K 2 , where K is R or C . Our goal is to study the singular point map S d , it sends polynomials f of degree d to their singular points 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 ; here both are polynomials of degree d, λ ∈ K ∗ . In order to describe the degree d essentially determined polynomials, a computation of the required number of isolated singular points δ (d) is provided. A dichotomy appears for the values of δ (d) ; depending on a certain parity, the space of essentially determined polynomials is an open or closed Zariski set. We compute the map 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 Aff (K 2) determines a singular K -analytic surface.
- Subjects
POLYNOMIALS; POLYNOMIAL rings; VECTOR fields
- Publication
Results in Mathematics / Resultate der Mathematik, 2024, Vol 79, Issue 3, p1
- ISSN
1422-6383
- Publication type
Article
- DOI
10.1007/s00025-024-02131-5