Invariant algebraic surfaces of polynomial vector fields in dimension three
Kruff, Niclas (Rheinisch-Westfälische Technische Hochschule Aachen. Lehrstuhl A für Mathematik)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Pantazi, Chara (Universitat Politècnica de Catalunya. Departament de Matemàtiques)
Walcher, Sebastian (Rheinisch-Westfälische Technische Hochschule Aachen. Lehrstuhl A für Mathematik)

Date:	2021
Abstract:	We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincaré from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi-invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi-invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria and properties for algebraic Jacobi multipliers. Some results are stated and proved for polynomial vector fields in arbitrary dimension and their invariant hypersurfaces. In dimension three we obtain detailed results on possible degree bounds. Moreover by an explicit construction we show for quadratic vector fields that the conditions involving the stationary points at infinity are generic but they do not a priori preclude the existence of invariant algebraic surfaces. In an appendix we prove a result on invariant lines of homogeneous polynomial vector fields.
Grants:	European Commission 777911 Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1617 Ministerio de Ciencia e Innovación MTM2016-77278-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1049 Ministerio de Economía y Competitividad MTM2015-65715-P Agencia Estatal de Investigación PGC2018-098676-B-I00
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Poincaré problem ; Darboux integrability ; Jacobi multiplier ; Nonassociative algebras
Published in:	Journal of dynamics and differential equations, (October 2021) , ISSN 1572-9222

DOI: 10.1007/s10884-021-10080-8

28 p, 467.4 KB

Postprint 23 p, 387.9 KB

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Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
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