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Pàgina inicial > Articles > Articles publicats > On the number of invariant conics for the polynomial vector fields defined on quadrics |
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. |
Ajuts: | Ministerio de Ciencia e Innovación MTM2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 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 ; Versió acceptada per publicar |
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 |
Postprint 29 p, 2.3 MB |