1. Let $X$ be a smooth affine algebraic variety. Does there necessarily exist an embedding into some affine space $A^n$ of codimension $1$?


I guess so. Next one I'm less sure.


2. Let $X$ be a complete intersection inside an affine space $A^m$. Does there exist a different embedding into another $A^n$ of codimension $1$?



Embedding varieties as divisors
Not every smooth affine curve can be embedded into the affine plane so the answer is no.


