Plane polynomial automorphisms of fixed multidegree

摘要

Let G be the group of polynomial automorphisms of the complex affine plane. On one hand, G can be endowed with the structure of an infinite dimensional algebraic group (see Shafarevich in Math USSR Izv 18:214-226, 1982) and on the other hand there is a partition of G according to the multidegree (see Friedland and Milnor in Ergod Th Dyn Syst 9: 67-99, 1989). Let G(d) denote the set of automorphisms whose multidegree is equal to d. We prove that G(d) is a smooth, locally closed subset of G and show some related results. We give some applications to the study of the varieties G(=m) (resp. G(<= m)) of automorphisms whose degree is equal to m (resp. is less than or equal to m).