Polynomial vector fields on the Clifford torus

TL;DR

This paper characterizes polynomial vector fields in four-dimensional space that leave the Clifford torus invariant and analyzes the maximum number of invariant meridians and parallels based on their degree.

Contribution

It provides a complete classification of polynomial vector fields with the Clifford torus as an invariant surface and investigates their invariant structures.

Findings

Characterization of all polynomial vector fields with the Clifford torus as an invariant surface

Determination of the maximum number of invariant meridians and parallels based on polynomial degree

Insights into the geometric structure of polynomial vector fields in four dimensions

Abstract

First we characterize all the polynomial vector fields in $\R^4$ which have the Clifford torus as an invariant surface. After we study the number of invariant meridians and parallels that such polynomial vector fields can have in function of the degree of these vector fields.