Assume that $\gamma$ is an analytic simple closed curve in $\mathbb{R}^2$ which surrounds origin.



> 
> Is there a polynomial vector field on the plane which is tangent to $\gamma$? In the other word, can an arbitrary analytic simple closed curve be realized as a closed orbit or a limit cycle of a polynomial vector field on the plane? 
> 
> 
> 



Polynomial vector field tangent to a given analytic simple closed curve
I see this as very unlikely. A polynomial vector field would have a slope function that is a rational function of two variables with a finite number of coefficients. 


As a consequence, if you take the field generated over $\mathbb Q$ by the coordinates $(x\_k, y\_k), k=1,\ldots, n$ of the points on the curve, as well as the slopes $c\_k$, this field will have transcendence degree at most $2n+{\rm constant}$. I see no reason for such bound to be true for an arbitrary closed analytic curve.


