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 = (f_n ⁿ≤1,forwhich f₁ = l, f₂ = ... = f_p+l =0,f_p+2 ≠ 0 for fixed, but arbitrary p ≤ 0 (see Theorem 5, Corollary 6 and Theorem 6). This representation uses a sequence (w ^p_n )_n≥p+2 of universal polynomials in f_p+2 and a sequence of parameters, which determines the individual one-parameter group. Instead of (w ^p_n )_n≥p+2 we may also use another sequence (L ^p_n )_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.