On the Variety of Automorphisms of the Affine Plane Original Research Article

The main subject of our study is GA2, n, the variety of automophisms of the affine plane of degree bounded by a positive integern. After detailing some definitions and notations in Section 1, we give in Section 2 an algorithm to decide whether an endomorphism of the affine plane over an integral domain is a tame automorphism. Then, by applying this algorithm to the Nagata automorphism, we recover easily its known results. In Section 6, we compute the number of irreducible components of GA2, nwhenn ≤ 9 and we show that GA2, nis reducible whenn ≥ 4. Our proofs are based on a precise decomposition theorem for automorphisms given in Section 3 and a characterization of length one automorphisms given in Section 5. Finally, in Section 7, we give some details on the casen = 4.