Let $\mathbb{A}^n\_k$ be the Affine $n$-space over an algebraically closed field $k$.
Let $X$ be a variety over $k$. What would be the right definition of an "Affine bundle" i.e bundle of fiber type $\mathbb{A}^n\_k$ over $X$ (I mean local triviality in zarisky topology,or
etale .. )?. When can one get a vector bundle from an "Affine bundle" , more
precisely (I think !) if I assume the structure group of the affine bundle to
be $Aut\_{Var\_k}(\mathbb{A}^n\_k)$, when can one get a reduction of the structure group
to $GL\_n(k)$?



Affine bundles over varieties
I think the term "affine bundle" is used for at least two things: (1) A map $p:Y\to X$ such that for some open cover (in your choice of topology) there are isomorphisms $p^{-1}(U)={\Bbb A}^n \times U$ --- just like you said. (2) A torsor for a vector bundle, i.e., like (1) but with the added condition that the transition functions are affine-linear.


In the situation of (2), it's a vector bundle exactly when there's a section (like any torsor). A simple non-vector-bundle-example is the complement of the diagonal in ${\Bbb P}^1 \times {\Bbb P}^1$, projecting onto one of the factors.


For (1), I don't know any general (non-trivial) criterion for such a thing to be a vector bundle. (Maybe because the group $Aut({\Bbb A}^n)$ is so complicated...) A simple non-example is the 2nd-order jet scheme $\mathrm{Hom}(\mathrm{Spec}(k[t]/(t^3)),{\Bbb P}^1) \to {\Bbb P}^1$. The fibers are ${\Bbb A}^2$, and there's a section, but it's not linear. I suppose one test is whether the sheaf of $O\_X$-algebras $p\_\*O\_Y$ admits a grading generated in degree one. (This fails for the jet schemes, though there is a natural grading by scaling $t$.)


