Roots, iterations and logarithms of formal automorphisms

In this paper it is proved that having a logarithm is e1uivalent to having roots of arbitrary order in the group of automorphisms of a formal power series ring, and in algebraic subgroups, too. INTRODUCTION The question whether an automorphism of a complex formal power series ring has an iteration, or may be embedded in a one parameter group has received much attention lately. The original setting concerned the possibility of embedding an automorphism in a complex analytic one dimensional Lie group (LEWIS [4, section 4] , STERNBERG [12, section 2]). In the recent terminology this is expressed as having a complex analytic iteration. At this moment the equivalence of the following statements has been proven: THEOREM 1 Let F be an automorphism of the complex formal power series ring £[[x 1 , ••• ,xm]].Then the following statements are equivalent: i F has a complex analytic iteration. ii F is conjugate to an automorphism in smooth normal form. iii F has a real continuous iteration. iv F has a rational continuous iteration. v F is the exponential of a derivation of c[ [xl"" ,x m ]] vi F has pairwise commuting roots of aU orders. vii F is conjugate to an automorphism in normal form which has roots of aU orders in normal form. The notions mentioned in this theorem are explained in the sequel. The proofs of the various equivalences may be found in: REICH SCHWAIGER [11, satz 4] for i ~ ii, BUCHER [1] for i ~ iii, PRAAGMAN [7, Theorem 5] for i ~ ii ~ iii ~ iv ~ v, PRAAGMAN [8, Theorem 3] for iv ~ V, MEHRING [5, satz 1.10] for i ~ vi, and PRAAGMAN [10, theorem 6] for V ~ vii. In the first section I shall prove that the condition pairwise commuting in vi may be deleted. In fact it will turn out that