Plane polynomials and Hamiltonian vector fields determined by their singular points

TL;DR

This paper investigates the relationship between plane polynomials and their critical points, characterizing when a polynomial is uniquely determined by its critical points, and explicitly describing such polynomials for degree three.

Contribution

It provides a detailed analysis of the critical point map for plane polynomials, computes the number of critical points needed for determination, and describes configurations for degree three polynomials.

Findings

Computed the critical point map for degree 3 polynomials.

Identified conditions under which polynomials are uniquely determined by their critical points.

Described explicit configurations for degree three polynomials with specific critical points.

Abstract

Let $\Sigma(f)$ be critical points of a polynomial $f \in \mathbb{K}[x,y]$ in the plane $\mathbb{K}^2$, where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$. Our goal is to study the critical point map $\mathfrak{S}_d$, by sending polynomials $f$ of degree $d$ to their critical points $\Sigma(f)$ . Very roughly speaking, a polynomial $f$ is essentially determined when any other $g$ sharing the critical points of $f$ satisfies that $f= \lambda g$; here both are polynomials of at most degree $d$, $\lambda \in \mathbb{K}^*$. In order to describe the degree $d$ essentially determined polynomials, a computation of the required number of isolated critical points $\delta (d)$ is provided. A dichotomy appears for the values of $\delta (d)$; depending on a certain parity the space of essentially determined polynomials is an open or closed Zariski set. We compute the map $\mathfrak{S}_3$, 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 $\hbox{Aff}(\mathbb{K}^2)$ determines a singular surface over $\mathbb{K}$.