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

Freudenburg, Gene

doi:10.1090/S0002-9939-06-08532-7

A linear counterexample to the Fourteenth Problem of Hilbert in dimension eleven
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Proc. Amer. Math. Soc. 135 (2007), 51-57 Request permission

Abstract:

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 $\mathbb {A}^{11}$.

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