Plane Polynomials and Hamiltonian Vector Fields Determined by Their Singular Points
Por:
Arredondo J.A., Muciño-Raymundo J.
Publicada:
1 ene 2024
Resumen:
Let S(f) be the singular points of a polynomial f?K[x,y] in the plane K2, where K is R or C. Our goal is to study the singular point map Sd, it sends polynomials f of degree d to their singular points S(f). Very roughly speaking, a polynomial f is essentially determined when any other g sharing the singular points of f satisfies that f=?g; here both are polynomials of degree d, ??K*. In order to describe the degree d essentially determined polynomials, a computation of the required number of isolated singular points d(d) is provided. A dichotomy appears for the values of d(d); depending on a certain parity, the space of essentially determined polynomials is an open or closed Zariski set. We compute the map S3, 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(K2) determines a singular K-analytic surface. © The Author(s) 2024.
Filiaciones:
Arredondo J.A.:
Fundación Universitaria Konrad Lorenz, Bogotá, 110231, Colombia
Muciño-Raymundo J.:
Centro de Ciencias Matemáticas, UNAM, MICH, Campus Morelia, Morelia, 58089, Mexico
Green Submitted, hybrid, All Open Access; Green Open Access; Hybrid Gold Open Access
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