Let $k$ be a field and $A$ be a finitely generated (commutative) algebra over $k$. If $A\_1$ and $A\_2$ are finitely generated $k$-subalgebras of $A$, is it true that $A\_1 \cap A\_2$ is also finitely generated (as an algebra) over $k$? What if $A$ is a polynomial ring?


**Update** (for the sake of completeness, April 1, 2017): [This paper](https://arxiv.org/abs/1301.2730) (disclaimer: it's mine) describes the smallest dimensional counterexample to the second question in zero characteristic. In positive characteristic the answer seems to be unknown.



Intersection of finitely generated subalgebras also finitely generated?
[Thomas Bayer](http://www.emis.de/journals/BAG/vol.43/no.2/19.html) has found a counter-example using rings of invariants inside polynomial rings.


