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, Wietzenbock 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, Wietzenbock, 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.

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.

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, Wietzenbock 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, Wietzenbock, lecture notes by Nagata which prove both theorems.

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, Wietzenbock 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, Wietzenbock, 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.