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Let $k=\overline k$ be an algebraically closed field (let's say characteristic 0 for simplicity.) I read the theorem that says

Every morphism $\varphi : \mathbb A_k^1 \to C$ (where $C \subseteq \mathbb A_k^n$ is an algebraic variety) is closed. (I.e. closed subsets of $\mathbb A_k^1$ get mapped to closed subsets of $C$.)

Unfortunately the proof was non constructive.

Given a concrete morphism $\varphi = (\varphi_1,\ldots,\varphi_n)$ where $\varphi_i \in k[X]$, is there an algorithm to find $p_i \in k[X_1,\ldots,X_n]$ such that $\varphi(\mathbb A_k^1) = V(p_1,\ldots,p_n)$?

The case $n=1$ is trivial. For general $n$ I think it might be too difficult, but can we say something about $n=2$?

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  • $\begingroup$ I think that would be the kernel of the map $k[X_1, \ldots, X_n] \to k[t]$ which maps $X_i$ to $\phi_i(t)$. The theory of Groebner bases provides a way to calculate kernels like this. $\endgroup$ – Daniel Schepler Apr 26 '17 at 19:42
  • $\begingroup$ Thank you for this hint. I'm not familiar with Groebner bases, can you recommend any references on that topic? $\endgroup$ – flawr Apr 26 '17 at 19:57
  • $\begingroup$ Personally I learned Groebner bases from Eisenbud's Commutative Algebra with a View Towards Algebraic Geometry. $\endgroup$ – Daniel Schepler Apr 26 '17 at 19:59

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