Finite vector bundles over punctured affine spaces Let $X$ be a connected scheme. Recall that a vector bundle $V$ on $X$ is called finite if there are two different polynomials $f,g \in \mathbb N[T]$ such that $f(V) = g(V)$ inside the semiring of vector bundles over $X$ (this definition is due to Nori, if I am not mistaken). For example, any trivial or torsion line bundle is finite in this sense.
Now, let $k$ be a field and $X= \mathbb A_k^n -\{0\}$ with $n\geq 3$. My question is:

Are there non-trivial finite vector bundles on $X$? If the answer is no, are there elementary ways to see this?

In the case $k=\mathbb C$, I think the answer is no, as  follows: by results of Nori in his thesis, finite bundles gives rise to representations of the fundamental groups scheme, so it is enough to see this group is trivial. But over $\mathbb C$, such group coincides with the etale fundamental group, and $X$ is simply connected. This argument seems to break down over arbitrary fields. 
Motivation: I would like to mention this in a talk next week! 
 A: If I understand your definition of "finite", then any stably trivial bundle is finite.  In particular, if R denotes the real numbers, the bundle over $Spec(R[X,Y,Z]/(X^2+Y^2+Z^2=1)$ defined by the unimodular row (X,Y,Z) is finite.  Unless I'm missing something, it should be easy to extend this example to the punctured affine space. $A_R^3-\{0\}$.
A: By Nori's theorem, the existence of non-trivial finite vector bundles on a reduced connected scheme $X$ of finite type over a perfect field $k$ with a rational point is equivalent to the fact that for every finite group scheme $G$ over $k$, every $G$-torsor is a pullback from $k$.
On $X = \mathbb A^n\smallsetminus \{0\}$ over an algebraically closed field of positive characteristic there are non-trivial finite bundles: in this case $X$ has lots of non-trivial connected étale covers (for example, by Artin-Schreier).
In characteristic 0, however, this does not happen; it follows from the equivalence between étale covers of $X$ and of $\mathop{\rm Spec}k$, and from the fact that every finite group scheme over $k$ is étale, that there is an equivalence of $G$-torsors on $X$ and on $\mathop{\rm Spec}k$.
[Edit]: Of course Torsten is right, Nori's theorem applies to projective varieties. The answer above is just wrong.
