Plane polynomials and Hamiltonian vector fields determined by their singular points release_k2wzwsewhrc3diuxkrenfvfedi

by John A. Arredondo, JesΓΊs MuciΓ±o-Raymundo

Released as a article .

2022  

Abstract

Let Ξ£(f) be critical points of a polynomial f βˆˆπ•‚[x,y] in the plane 𝕂^2, where 𝕂 is ℝ or β„‚. Our goal is to study the critical point map 𝔖_d, by sending polynomials f of degree d to their critical points Ξ£(f) . Very roughly speaking, a polynomial f is essentially determined when any other g sharing the critical points of f satisfies that f= Ξ» g; here both are polynomials of at most degree d, Ξ»βˆˆπ•‚^*. In order to describe the degree d essentially determined polynomials, a computation of the required number of isolated critical points Ξ΄ (d) is provided. A dichotomy appears for the values of Ξ΄ (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 four points leads to a degree three essentially determined polynomial. Furthermore, we describe explicitly configurations supporting degree three non essential determined polynomials. The quotient space of essentially determined polynomials of degree three up to the action of the affine group Aff(𝕂^2) determines a singular surface over 𝕂.
In text/plain format

Archived Files and Locations

application/pdf  1.7 MB
file_bnurlyo2prdk3hq6fblwhky3j4
arxiv.org (repository)
web.archive.org (webarchive)
Read Archived PDF
Preserved and Accessible
Type  article
Stage   submitted
Date   2022-06-11
Version   v1
Language   en ?
arXiv  2206.05569v1
Work Entity
access all versions, variants, and formats of this works (eg, pre-prints)
Catalog Record
Revision: 0fd9ce88-e685-4b3a-a6b6-82ad05f8096d
API URL: JSON