Recently, Shestakov-Umirbaev solved Nagata’s conjecture on an automorphism of a polynomial ring. To solve the conjecture, they defined notions called reductions of types I–IV for automorphisms of a polynomial ring. An automorphism admitting a reduction of type I was first found by Shestakov-Umirbaev. Using a computer, van den Essen–Makar-Limanov–Willems gave a family of such automorphisms. In this paper, we present a new construction of such automorphisms using locally nilpotent derivations. As a consequence, we discover that there exists an automorphism admitting a reduction of type I which satisfies some degree condition for each possible value.
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Shigeru Kuroda, Automorphisms of a Polynomial Ring Which Admit Reductions of Type I. Publ. Res. Inst. Math. Sci. 45 (2009), no. 3, pp. 907–917DOI 10.2977/PRIMS/1249478968