Wild multidegrees of the form \((d,d_{2},d_{3})\) for fixed \(d\geq 3\) (Q2903921)

Let \(\text{Aut}(\mathbb{C}^{n})\) be the group of polynomial automorphisms of \(\mathbb{C}^{n}\) and \(\text{Tame}(\mathbb{C}^{n})\subset \text{Aut}( \mathbb{C}^{n})\) be the subgroup of tame automorphisms which are compositions of linear and triangular automorphisms. The automorphisms in \(\text{Aut}( \mathbb{C}^{n})\setminus \text{Tame}(\mathbb{C}^{n})\) are called wild. The famous Nagata example is a wild automorphism. If \(F=(f_{1},\ldots ,f_{n})\in \text{Aut}(\mathbb{C}^{n})\) then \(\text{mdeg}(F):=(\deg f_{1},\ldots ,\deg f_{n})\) is the multidegree of \(F.\) The authors, using the Nagata example and a characterization of multidegres of tame automorphisms (given by the first author) prove that for \(n=3\) and any \(d>2\) the set NEWLINE\[NEWLINE \{(d_{1},d_{2},d_{3})\in \text{mdeg}(\text{Aut}(\mathbb{C} ^{n}))\setminus \text{mdeg}(\text{Tame}(\mathbb{C}^{n})):d=d_{1}\leq d_{2}\leq d_{3}\} NEWLINE\]NEWLINE is infinite.