Darboux theory of integrability for polynomial vector fields on 𝕊ⁿ

ABSTRACT

This is a survey on the Darboux theory of integrability for polynomial vector fields, first in and second in the n-dimensional sphere . We also provide new results about the maximum number of invariant parallels and meridians that a polynomial vector field on can have in function of its degree. These results in some sense extend the known result on the maximum number of hyperplanes that a polynomial vector field in can have in function of the degree of .

KEYWORDS:

• Darboux integrability theory
• invariant meridians
• invariant parallels
• n-dimensional sphere

MATHEMATICS SUBJECT CLASSIFICATION CODES:

• Primary 34C07
• 34C05
• 34C40

Acknowledgments

The first author is partially supported by a FEDER-MINECO grant MTM2016-77278-P, a MINECO grant MTM2013-40998-P, and an AGAUR grant 2014SGR-568. The second author acknowledges a BITDEFENDER postdoctoral fellowship from the Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Contract of Sponsorship No. 262/2016 as well as partial support from a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-0657.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

FEDER-MINECO [grant number MTM2016-77278-P], MINECO [grant number MTM2013-40998-P], AGAUR [grant number 2014SGR-568]; Institute of Mathematics “Simion Stoilow” of the Romanian Academy [contract of sponsorship number 262/2016]; Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI [project number PN-II-RU-TE-2014-4-0657].