Separability of wild automorphisms of a polynomial ring
Let \(R[{\mathbf x}]=R[x_1,\dots,x_n]\) be the polynomial algebra in \(n\) variables over a domain \(R\) and let \(\Gamma\) be a totally ordered additive group. For an \(n\)-tuple \({\mathbf w}=(w_1,\dots,w_n)\in\Gamma^n\) one defines a \({\mathbf w}\)-weighted degree of the polynomials in \(R[{\mathbf x}]\) assuming that \(\text{deg}_{\mathbf w}(x_i)=w_i\). Then, for every automorphism \(F=(f_1,\dots,f_n)\) of \(R[{\mathbf x}]\) one defines the \({\mathbf w}\)-weighted multidegree of \(F\) as \(\text{mdeg}_{\mathbf w}(F)=(\text{deg}_{\mathbf w}(f_1),\dots,\text{deg}_{\mathbf w}(f_n))\). In the paper under review, the authors study the \({\mathbf w}\)-weighted multidegree of the automorphisms of \(k[x,y,z]\) over a field \(k\) of characteristic 0 for \(\Gamma={\mathbb Z}\). The main result gives that for a large class of wild automorphisms \(F\) of \(k[x,y,z]\) one can find a weight \(\mathbf w\) such that there are no tame automorphisms with the same \({\mathbf w}\)-weight multidegree \(\text{mdeg}_{\mathbf w}(F)\). For example, if \({\mathbf w}=(5,4,3)\), then for the Nagata automorphism \(N\) the weighted \(\mathbf w\)-multidegree of \(N\) is \(\text{mdeg}_{\mathbf w}(N)=(19,11,3)\) and there are no tame automorphisms with the same weighted \(\mathbf w\)-multidegree. (If \({\mathbf w}=(1,1,1)\), then the Nagata automorphism and the tame automorphism \((x+z^5,y+z^3,z)\) have the same multidegree \((5,3,1)\).)
scientific article; zbMATH DE number 6160923
scientific article