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On the number of invariant conics for the polynomial vector fields defined on quadrics
Bolaños Rivera, Yudi Marcela (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Valls, Clàudia (Universidade Técnica de Lisboa. Departamento de Matemática)

 Data: 2013 Resum: The quadrics here considered are the nine real quadrics: parabolic cylinder, elliptic cylinder, hyperbolic cylinder, cone, hyperboloid of one sheet, hyperbolic paraboloid, elliptic paraboloid, ellipsoid and hyperboloid of two sheets. Let Q be one of these quadrics. We consider a polynomial vector field X = (P, Q, R) in R3 whose flow leaves Q invariant. If m1 = degree P, m2 = degree Q and m3 = degree R, we say that m = (m1, m2, m3) is the degree of X. In function of these degrees we find a bound for the maximum number of invariant conics of X that result from the intersection of invariant planes of X with Q. The conics obtained can be degenerate or not. Since the first six quadrics mentioned are ruled surfaces, the degenerate conics obtained are formed by a point, a double straight line, two parallel straight lines, or two intersecting straight lines; thus for the vector fields defined on these quadrics we get a bound for the maximum number of invariant straight lines contained in invariant planes of X. In the same way, if the conic is non degenerate, it can be a parabola, an ellipse or a hyperbola and we provide a bound for the maximum number of invariant non degenerate conics of the vector field X depending on each quadric Q and of the degrees m1, m2 and m3 of X. Nota: Número d'acord de subvenció MICIIN/MTM2008-03437 Nota: Número d'acord de subvenció AGAUR/2014/SGR-410 Nota: Agraïments: The third author is supported by the grants AGAUR PIV-DGR-2010 and by FCT through CAMGD Drets: Tots els drets reservats. Llengua: Anglès. Document: article ; recerca ; submittedVersion Matèria: polynomial vector fields ; invariant quadrics ; invariant conics ; extactic polynomial Publicat a: Bulletin des Sciences Mathematiques, Vol. 137 (2013) , p. 746-774, ISSN 0007-4497

DOI: 10.1016/j.bulsci.2013.04.003

 Preprint 29 p, 2.3 MB

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