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!