Derivations and polynomial automorphisms

In Chapter 1 we encountered several polynomial automorphisms, in particular the elementary ones and the Nagata example. In § 1 of this chapter, we describe a uniform way to construct both examples, namely as so-called exponential automorphisms, i.e. automorphisms of the form exp D where D is a locally nilpotent derivation on the polynomial ring k [ X ], see Examples 2.1.9 and 2.1.10. Since in case R is not a field there exist non-tame automorphisms of R [ X ] of the form exp D , this will be proved in Chapter 5, it is natural to replace the tame generators conjecture by a weaker one, the so-called exponential generators conjecture, which asserts that for any ℚ-algebra R , every R -automorphism of R [ X ] is a finite product of exponential automorphisms and elements of Aff (R , n). In §1 we prove the exponential conjecture for the so-called nilpotency subgroup of Aut R R [ X ], i.e. the set of all automorphisms of the form ( X 1 + g 1 , …, X n + g n ), where each g i is a nilpotent element of R [ X ]. In fact we show that each such automorphism is an exponential automorphism! These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.