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 $n=2$?