Feuilletages holomorphes de codimension 1: une étude locale dans le cas dicritique

Cerveau, Dominique; Neto, Alcides Lins; Ravara-Vago, Marianna

doi:10.24033/asens.2280

Cited by 2 publications

(2 citation statements)

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“…It is known that no general degree bounds can exist for algebraic solutions, and that certain properties of singular points are relevant for establishing degree bounds in some classes. Considering dimension n ≥ 3, much work has been done in the last two decades to classify and characterize invariant surfaces; see for instance the survey [6] by Cerveau, and the work [8] by Cerveau et al on local properties. Further notable contributions concerning invariant algebraic varieties are due to Brunella and Gustavo Mendes [3], Cavalier and Lehmann [5], Corrêa and da Silva Machado [11,12], Corrêa and Jardim [10], Corrêa and Soares [13], Esteves [16], and Soares [29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

et al. 2021

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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.

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Section: Introductionmentioning

confidence: 99%

Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

et al. 2021

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“…For higher dimensions Jouanolou [10] showed the existence of the general degree bound m + 1 for semi-invariants of system (1) that define smooth hypersurfaces in projective space, and Soares [21,22,23] extended and refined this result. For dimension n ≥ 3 much work has been done in the last decade to classify and characterize invariant surfaces; see for instance the survey [4] by Cerveau, and the work [6] by Cerveau et al on…”

Section: Introductionmentioning

confidence: 99%