Commuting planar polynomial vector fields for conservative Newton systems

TL;DR

This paper characterizes polynomial vector fields commuting with a given planar polynomial vector field, revealing that the module of such derivations is of rank 1 when the degree of the polynomial is at least 2, impacting the understanding of linearizability.

Contribution

It proves that the module of polynomial derivations commuting with a given polynomial vector field has rank 1 if and only if the polynomial degree is at least 2, providing a new criterion for linearizability.

Findings

The module of commuting polynomial derivations is of rank 1 for degree ≥ 2.

Classical elliptic equations like $ ext{d}^2x=6x^2+a$ are included in this category.

The result links polynomial degree to the structure of commuting vector fields.

Abstract

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. Let $f \in K[x]$, where $K$ is a field of characteristic zero, and $d$ the derivation that corresponds to the differential equation $\ddot x = f(x)$ in a standard way. Let also $H$ be the Hamiltonian polynomial for $d$, that is $H=\frac{1}{2}y^2-\int{f(x)dx}$. It is known that the set of all polynomial derivations that commute with $d$ forms a $K[H]$-module $M_d$. In this paper, we show that, for every such $d$, the module $M_d$ is of rank $1$ if and only if $\text{deg}\; f\geqslant 2$. For example, the classical elliptic equation $\ddot x = 6x^2+a$, where $a \in \mathbb{C}$, falls into this category.