On the number of invariant conics for the polynomial vector fields defined on quadrics [ Back ]

15:30 to 16:30
UAB - Dept. Matemàtiques (C1/-128)
Yudy Bolaños
Universitat Autònoma de Barcelona


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 $C$ be one of these quadrics. We consider a polynomial vector field $X=(P,Q,R)$ in $\mathbb{R}^3$ whose flow leaves $C$ invariant. If $m_1= $degree$P$, $m_2= $degree $Q$ and $m_3= $degree$R$, we say that $m=(m_1,m_2,m_3)$ 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 $C.$ 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 $C$ and of the degrees $m_1$, $m_2$ and $m_3$ of $X.$

This work is in collaboration with J. Llibre and C. Valls.