scholarly journals Holomorphic foliations in ${\Bbb C}{\Bbb P}(2)$ having an invariant algebraic curve

1991 ◽  
Vol 41 (4) ◽  
pp. 883-903 ◽  
Author(s):  
Dominique Cerveau ◽  
Alcides Lins Neto
Keyword(s):  
Algebraic Curve ◽  
Holomorphic Foliations ◽  
Invariant Algebraic Curve

scholarly journals Finitely curved orbits of complex polynomial vector fields

2007 ◽  
Vol 79 (1) ◽  
pp. 13-16
Author(s):  
Albetã C. Mafra
Keyword(s):  
Vector Field ◽  
Gaussian Curvature ◽  
Algebraic Curve ◽  
Vector Fields ◽  
Complex Polynomial ◽  
Holomorphic Foliations ◽  
Holomorphic Curve ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

This note is about the geometry of holomorphic foliations. Let X be a polynomial vector field with isolated singularities on C². We announce some results regarding two problems: 1. Given a finitely curved orbit L of X, under which conditions is L algebraic? 2. If X has some non-algebraic finitely curved orbit L what is the classification of X? Problem 1 is related to the following question: Let C <FONT FACE=Symbol>Ì</FONT> C² be a holomorphic curve which has finite total Gaussian curvature. IsC contained in an algebraic curve?

AN ALGORITHM FOR FINDING INVARIANT ALGEBRAIC CURVES OF A GIVEN DEGREE FOR POLYNOMIAL PLANAR VECTOR FIELDS

2005 ◽  
Vol 15 (03) ◽  
pp. 1033-1044 ◽  
Author(s):  
GRZEGORZ ŚWIRSZCZ
Keyword(s):  
Differential Equations ◽  
Algebraic Curve ◽  
Vector Fields ◽  
Algebraic Curves ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Invariant Algebraic Curves ◽  
Invariant Algebraic Curve ◽  
Planar Vector Fields ◽  
Planar Vector

Given a system of two autonomous ordinary differential equations whose right-hand sides are polynomials, it is very hard to tell if any nonsingular trajectories of the system are contained in algebraic curves. We present an effective method of deciding whether a given system has an invariant algebraic curve of a given degree. The method also allows the construction of examples of polynomial systems with invariant algebraic curves of a given degree. We present the first known example of a degree 6 algebraic saddle-loop for polynomial system of degree 2, which has been found using the described method. We also present some new examples of invariant algebraic curves of degrees 4 and 5 with an interesting geometry.

scholarly journals Liouvillian first integrals for a class of generalized Liénard polynomial differential systems

2016 ◽  
Vol 146 (6) ◽  
pp. 1195-1210 ◽  
Author(s):  
Jaume Llibre ◽  
Clàudia Valls
Keyword(s):  
Algebraic Curve ◽  
First Integrals ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Invariant Algebraic Curve

We study the existence of Liouvillian first integrals for the generalized Liénard polynomial differential systems of the form xʹ = y, yʹ = –g(x) – f(x)y, where f(x) = 3Q(x)Qʹ(x)P(x) + Q(x)2Pʹ(x) and g(x) = Q(x)Qʹ(x)(Q(x)2P(x)2 – 1) with P,Q ∈ ℂ[x]. This class of generalized Liénard polynomial differential systems has the invariant algebraic curve (y + Q(x)P(x))2 – Q(x)2 = 0 of hyperelliptic type.

scholarly journals Algebraic Limit Cycles on Quadratic Polynomial Differential Systems

2018 ◽  
Vol 61 (2) ◽  
pp. 499-512 ◽  
Author(s):  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Algebraic Curve ◽  
Positive Answer ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Invariant Algebraic Curve ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

AbstractAlgebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and a few years later the following conjecture appeared: quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that a quadratic polynomial differential system having an invariant algebraic curve with at most one pair of diametrically opposite singular points at infinity has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.

scholarly journals Existence of dicritical singularities of Levi-flat hypersurfaces and holomorphic foliations

2017 ◽  
Vol 196 (1) ◽  
pp. 35-44 ◽  
Author(s):  
Andrés Beltrán ◽  
Arturo Fernández-Pérez ◽  
Hernán Neciosup
Keyword(s):  
Holomorphic Foliations ◽  
Dicritical Singularities

On multiple curves. II

1945 ◽  
Vol 41 (2) ◽  
pp. 117-126
Author(s):  
W. V. D. Hodge
Keyword(s):  
Algebraic Curve ◽  
Abelian Integrals

In this note I consider the Abelian integrals of the first kind on an algebraic curve Γ which is a normal multiple of a curve C, as defined in Note I*.

scholarly journals On Characteristic Forms of Holomorphic Foliations

1999 ◽  
Vol 22 (1) ◽  
pp. 43-64
Author(s):  
Yoshikatsu KAMOZAWA ◽  
Masahide KATO
Keyword(s):  
Holomorphic Foliations
Sign in / Sign up

Export Citation Format

Share Document