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Question: When a quadratic vector field $V$ does not have a center on the plane, is the curve $$\{(x,y)\mid \kappa(x,y)=0\}$$ transverse to $V$?

 

If not, what appropriate rescaling of the the second vector $ \frac{1}{x^2+y^2} W$ of the orthonormal frame would give a positive answer?

Question: When a quadratic vector field $V$ does not have a center on the plane, is the curve $$\{(x,y)\mid \kappa(x,y)=0\}$$ transverse to $V$?

If not, what appropriate rescaling of the the second vector $ \frac{1}{x^2+y^2} W$ of the orthonormal frame would give a positive answer?

If not, what rescaling of the the second vector $ \frac{1}{x^2+y^2} W$ of the orthonormal frame would give a positive answer?

If not, what appropriate rescaling of the the second vector $ \frac{1}{x^2+y^2} W$ of the orthonormal frame would give a positive answer?

(Hilbert 16th problem for n=2) IS $H(2)$ a finite number?

where $V=P\partial_x +Q\partial_y$ is the quadratic vector field as in $(V)$ and $W$ is the radial vector field $W=x\partial_x+y\partial_y$. In fact $W$, as it is required in the proof of the above two propositions, lies in the kernel of $1\_$ form $\psi=\frac{1}{x^2+y^2}(ydx-xdy)$. This $1\_$ form is very essential to apply propositions 6.7 and 6.8 in the above reference. In fact this is a closed form which is identically equal to $1$ on the first vector of the orthonormal frame.That is $\psi(\frac{x^2+y^2}{yP-xQ}(ydx-xdy))=1$. I am indebted to Ben McKay who suggested the $1\_$form $d\theta$ as a required $1\_$ form for possible satisfactions of proposition 6.8. On the other hand the method of the proof of the Proposition 6.7 shows that, on the complement of $C$, all trajectories of $V$ are geodesics for the metric arising from the above orthonormal frame. Moreover we are free to rescale the second vector of the frame, arbitrarily.

(Hilbert 16th problem for n=2) Is $H(2)$ a finite number?

where $V=P\partial_x +Q\partial_y$ is the quadratic vector field as in $(V)$ and $W$ is the radial vector field $W=x\partial_x+y\partial_y$. In fact $W$, as it is required in the proof of the above two propositions, lies in the kernel of $1\_$ form $\psi=\frac{1}{x^2+y^2}(ydx-xdy)$. This $1\_$ form is very essential to apply propositions 6.7 and 6.8 in the above reference. In fact this is a closed form which is identically equal to $1$ on the first vector of the above orthonormal frame.That is $\psi(\frac{x^2+y^2}{yP-xQ}(V))=1$. 

I am really indebted to Ben McKay who suggested the $1\_$form $d\theta$ as a required $1\_$ form for possible satisfactions of proposition 6.8.

On the other hand the method of the proof of the Proposition 6.7 shows that, on the complement of $C$, all trajectories of $V$ are geodesics for the metric arising from the above orthonormal frame. Moreover we are free to rescale the second vector of the frame, arbitrarily.