On the solutions of the translation equation in rings of formal power series

Abstract

We study in this paper solutions of the translation equation in rings of formal power series K[X] where K ∈R, C (so called one-parameter groups or flows), and even, more generally, homomorphisms Ф from an abelian group (G, +) into the group Г(K) of invertible power series in K[X]. This problem can equivalently be formulated as the question of constructing homomorphisms Ф from (G, +) into the differential group Г1∞ describing the chain rules of higher order of C∞ functions with fixed point 0. In this paper we present the general form of these homomorphisms Ф : G → Г(K) (or L1∞),Ф = (fn n≤1,forwhich f1 = l, f2 = ... = fp+l =0,fp+2 ≠ 0 for fixed, but arbitrary p ≤ 0 (see Theorem 5, Corollary 6 and Theorem 6). This representation uses a sequence (w n p )n≥p+2 of universal polynomials in fp+2 and a sequence of parameters, which determines the individual one-parameter group. Instead of (w n p )n≥p+2 we may also use another sequence (L n p )n≥p+2 of universal polynomials, and we describe the connection between these forms of the solutions. The problem for homomorphisms Ф with f1≠1 will be treated in a separate paper.