On some reductive group actions on affine space. II (Q1191443)

[For part I see Group actions and invariant theory, Proc. Conf., Montreal/Can. 1988, CMS Conf. Proc. 10, 67-72 (1989; Zbl 0701.14041).] Let \(G\) be a linearly reductive group over an algebraically closed field \(k\), and \(\alpha:G\times V\to V\) be an algebraic action of \(G\) on a finite dimensional vector space \(V\). In part I (loc. cit.) it is proved, assuming \(\text{char} k=0\), that \(\alpha\) can be linearized under the following condition: \((*)\) There exists an integer \(d\geq 2\) such that the action has the form: \(g\mapsto\alpha(g)=\lambda(g)+\theta(g)\in\text{Mor}(V,V)\), for all \(g\in G\), where \(\lambda(g)\) is a linear map and \(\theta(g)=\theta_ d(g)+\theta_{d+1}(g)+\cdots+\theta_{2d-2}(g)\), where \(\theta_ i\) is homogeneous of degree \(i\), \(d\leq i\leq 2d-2\). The main goal of the present paper is to remove the restriction on the characteristic of \(k\), by proving that the explicit formula for a linearizing automorphism \(\sigma_ \alpha\) works in any characteristic. In the final remark the author also concludes that within some frames, even without assuming \((*)\), \(\sigma_ \alpha\) is ``in general'' linearizing for a \({\mathcal C}^*\) action \(\alpha\) on an affine space.