On the number of invariant conics for the polynomial vector fields defined on quadrics
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 fi...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2013 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:150629 |
| Acceso en línea: | https://ddd.uab.cat/record/150629 https://dx.doi.org/urn:doi:10.1016/j.bulsci.2013.04.003 |
| Access Level: | acceso abierto |
| Palabra clave: | Polynomial vector fields Invariant quadrics Invariant conics Extactic polynomial |
| Sumario: | 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. |
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