Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three
Journal of Dynamics and Differential Equations, vol. 35, pp. 3241–3268
Abstract
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.
Authors 4
-
Affiliation as printed
Lehrstuhl A für Mathematik, RWTH Aachen, Aachen, 52056, Germany
-
Universitat Autònoma de Barcelona
Affiliation as printed
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C Bellaterra, 08193, Barcelona, Catalonia, Spain
-
Chara Pantazi corresponding
Universitat Politècnica de Catalunya
Affiliation as printed
Departament de Matemàtiques, Universita Politècnica de Catalunya, (EPSEB), Av. Doctor Marañón, 44–50, 08028, Barcelona, Catalonia, Spain
-
Affiliation as printed
Lehrstuhl A für Mathematik, RWTH Aachen, Aachen, 52056, Germany
Cited by 2 stored of 2
2 results
No patents citing this paper on Lens.org (checked 2026-10-06).