A family of G_a-actions on affine space A~m is constructed, each having a non-finitely generated ring of invariants (m≥6). Because these actions are of small degree, they induce linear actions of unipotent groups G~n_a *G_a on A~(2n+3) for n≥4, and these invariant rings are also non-finitely generated. The smallest such action presented here is for the group G~n_4 *G_a acting linearly on A~(11).
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