Assume that we have an analytic simple closed curve \gamma in the plane.
Is there a polynomial vector field which is tangent to \gamma?
I asked this question at MO, too.
https://mathoverflow.net/questions/282139/polynomial-vector-field-tangent-to-a-given-analytic-simple-closed-curve
Dear Ali, if for you an approximation is sufficient, being the curve Gamma analytic (=differentiable many times?), formally you can use a Taylor (locally but it could be globally too) expansion in x and y coordinates of Gamma, and then derive it to have a polynomial vector field. Gianluca
Dear Gianluca
I thank you for your answer.
It is interesting and helpfull from the virw of approximation theory.
But in my question I search for an exact (but not approximated) solution.
Best regards
Ali
Ali, if gamma curve is C1-differentiable and defined on a compact subset of R2, then from Weierstrass Theorem it is expressed by a series of polynomials, and the same is for its tangent field (that is C0). But in general, for a smooth curve the tangent field is not expressed by a finite number of polynomials. Gianluca
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