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Let $\phi: \mathbb{R}^1 \longrightarrow \mathbb{R}^2$ be the map given by $t \mapsto (t^2,t^3)$. I'm trying to show that any polynomial $f \in \mathbb{R}[X,Y]$ vanishing on the image $C = \phi(\mathbb{R}^1)$ is divisible by $Y^2-X^3$. And what property of a field $k$ will ensure that the result holds for $\phi: k \longrightarrow k^2$ given by the same formula?

Also, I am trying to do it for $t \mapsto (t^2-1,t^3-t)$.

Thanks.

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    Dear Mary, The necessary and sufficient criterion is that $k$ should be infinite. Regards,2012-09-10
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    Can you please explain that and try to give another proof of this question. What Rod has looks good, but isn't what I'm looking for2012-09-10
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    Dear Mary, Georges has elaborated on my comment in his answer below. Best wishes,2012-09-11
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    P.S. If you want to see whether or not you understand his technique, you could try applying it to a similar but different map, such as $t \mapsto (t^2, t^5)$.2012-09-11

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