Actions of \(G_a\) on \(\mathbb{A}^3\) defined by homogeneous derivations (Q1380066)

Let \(k\) be a field. The symbol \(\mathbb{G}_a\) denotes the additive group \((k,+)\). In the present article the author studies an action of \(\mathbb{G}_a\) on the affine \(n\)-space \(\mathbb{A}^n\). In J. Pure Appl. Algebra 105, No. 3, 267-275 (1995; Zbl 0857.14024), \textit{G. Freudenburg} defined the rank of such an action. It has the following properties: (1) \(0\leq \text{rank}\leq n\); (2) an action of \(\mathbb{G}_a\) on \(\mathbb{A}^n\) of rank \(r\) can be embedded into one of \(\mathbb{G}_a\) on \(\mathbb{A}^{n'}\) of rank \(n'-n+r\) if \(n'>n\). If \(n=2\), then the rank is at most 1. This article gives an action of \(\mathbb{G}_a\) on \(\mathbb{A}^n\) of rank \(n\) for the first time if \(n\geq 3\).