A linear counterexample to the Fourteenth Problem of Hilbert in dimension eleven

A family of $\mathbb{G}_a$-actions on affine space $\mathbb{A}^m$ is constructed, each having a non-finitely generated ring of invariants ($m\ge 6$). Because these actions are of small degree, they induce linear actions of unipotent groups $\mathbb{G}_a^n\rtimes\mathbb{G}_a$ on $\mathbb{A}^{2n+3}$ for $n\ge 4$, and these invariant rings are also non-finitely generated. The smallest such action presented here is for the group $\mathbb{G}_a^4\rtimes\mathbb{G}_a$ acting linearly on $\mathbb{A}^{11}$.