Dynamics and integrability of polynomial vector fields on the $n$-dimensional sphere

TL;DR

This paper characterizes polynomial vector fields on spheres, establishing conditions for Hamiltonian structure, classifying fields with invariant spheres, and determining bounds for invariant hyperplanes, advancing understanding of their integrability and invariants.

Contribution

It provides new classifications, bounds, and characterizations of polynomial vector fields on spheres, including Hamiltonian conditions and invariant structures, which were previously not fully understood.

Findings

Necessary and sufficient condition for degree one Hamiltonian vector fields on odd-dimensional spheres.

Classification of degree two polynomial vector fields with invariant great spheres.

Sharp bounds for the number of invariant hyperplanes on spheres.

Abstract

In this paper, we characterize arbitrary polynomial vector fields on $S^n$. We establish a necessary and sufficient condition for a degree one vector field on the odd-dimensional sphere $S^{2n-1}$ to be Hamiltonian. Additionally, we classify polynomial vector fields on $S^n$ up to degree two that possess an invariant great $(n-1)$-sphere. We present a class of completely integrable vector fields on $S^n$. We found a sharp bound for the number of invariant meridian hyperplanes for a polynomial vector field on $S^2$. Furthermore, we compute the sharp bound for the number of invariant parallel hyperplanes for any polynomial vector field on $S^n$. Finally, we study homogeneous polynomial vector fields on $S^n$, providing a characterization of their invariant $(n-1)$-spheres.