Regular \(G_a\) invariants.
Let \(\delta\) be a locally nilpotent derivation on \(\mathbb C[x_1,\dots,x_n]\) such that the associated \(G_a\)-action is fixed point free. Suppose that the ring of invariants \(R := \mathbb C[x_1,\dots,x_n]^{\delta}\) is finitely generated. Let \(\mathfrak m \subset R\) be a maximal ideal such that \(R_{\mathfrak m}\) is a regular local ring and \(S := R \smallsetminus \mathfrak m\). It is proved that \(\Omega_{S^{-1}\mathbb C[x_1,\dots,x_n]| R_{\mathfrak m}}\) is free of rank one. A consequence is the following result: If \(R\) is regular then the associated \(G_a\)-action is locally trivial if and only if it is proper. Finally it is proved that a set of polynomials \(\{f_1,\dots,f_{n-1}\}\) is part of a coordinate system for \(\mathbb C^n\) (i.e. there exist \(f_n\) such that \(\mathbb C[x_1,\dots,x_n] = \mathbb C[f_1,\dots,f_n]\)) if and only if \(\mathbb C[f_1,\dots,f_{n-1}]\) is the ring of invariants for a proper \(G_a\)-action (the action is generated by the derivation \(\delta\) defined by \(\delta(h) = \lambda \det(J(f_1,\dots,f_{n-1},h))\) for some \(\lambda \in \mathbb C^*\)).
scientific article; zbMATH DE number 1799572