Let $R$ be a finitely generated commutative ring over a field, for concreteness.
If $S,T \leq R$ are two finitely generated subrings, is their intersection also finitely generated?
(Certainly this isn't true for infinite intersections.)
If so,
How can one compute such an intersection using a computer algebra package?
It's easy to reduce to the case that $R = {\mathrm k}[s_1,\ldots,s_n,t_1,\ldots,t_m]/I$, the subring $S$ is generated by $s_1,\ldots,s_n$, and $T$ is generated by $t_1,\ldots,t_m$.
No. Nagata gave an example of an linear action of $\mathbb{G}_a^k$ on a complex vector space $V$ such the ring of invariants $\mathrm{Sym}^{\bullet}(V)^{\mathbb{G}_a^k}$ is not finitely generated. On the other hand, Weitzenbock proved that, if $\mathbb{G}_a$ acts linearly on a complex vector space, then $\mathrm{Sym}^{\bullet}(V)^{\mathbb{G}_a}$ is finitely generated. The invariants for $\mathbb{G}_a^k$ can be written as the intersection of the invariants for $k$ different actions of $\mathbb{G}_a$.
References: Nagata, Weitzenbock, lecture notes by Nagata which prove both theorems.
I can give you an explicit example. Nagata starts with $\mathbb{G}_a^n$ acting on $k[x_1, \ldots, x_n, y_1, \ldots, y_n]$ by fixing the $x_i$ and having $(t_1, \ldots, t_n) \cdot y_j = y_j + t_j x_j$. He then proves that, for a carefully chosen vector subspace $G \subset \mathbb{G}_a^n$, the invariants $k[x_1, \ldots, x_n, y_1, \ldots, y_n]^G$ are not finitely generated.
Set $z_j=y_j/x_j$. So $k[x_1, \ldots, x_n, y_1, \ldots, y_n]$ embeds in $k[x_1, \ldots, x_n, z_1, \ldots, z_n]$, and $G$ acts on the $z$'s by translations. So the invariant ring $k[x_1, \ldots, x_n, z_1, \ldots, z_n]^G$ is a polynomial ring in $2n-\mathrm{dim}\ G$ variables.
We see that $k[x_1, \ldots, x_n, y_1, \ldots, y_n]$ and $k[x_1, \ldots, x_n, z_1,\ldots, z_n]^G$ are both polynomial rings, but their intersection is not finitely generated.
A couple of years ago, David Speyer showed me the following counterexample.
Let $E$ be an elliptic curve defined over a field, with points $P$, $Q$ that are linearly independent under the group law. Let $p:X\to E$ be the total space of the rank two vector bundle $\mathcal{O}([P])\oplus\mathcal{O}(-[Q])$ on $E$. Then $X$ is finite type, but it can be checked that the ring $\Gamma(\mathcal{O}_X)$ of global regular functions on $X$ is not finitely generated.
If we pick an affine open cover $\{U,V\}$ of $E$, then $\{p^{-1}(U),p^{-1}(V)\}$ is an affine open cover of $X$. The rings of regular functions on $p^{-1}(U)$ and $p^{-1}(V)$ are finitely generated, but their intersection inside the function field of $X$ (or inside the finitely generated subalgebra they generate) is $\Gamma(\mathcal{O}_X)$, which is not finitely generated.
