On the form of homomorphisms into the differential group L_s ¹ and their extensibility

Abstract

Let L_s ¹ (s ∈ ℕ) be the s-th differential group, that is the set {(x1,…,x_s): x₁ ≠ 0, x_n ∈ K, n =1,2,…,s} (K ∈ {ℝ,ℂ}) together with the group operation which describes the chain rules (up to order s) for C^s-functions with fixed point 0. We consider homomorphisms Φ_s, Φ_s = (f₁,…,f_s) from an abelian group (G,+) into L_s ¹ such that f₁ = 1, f₂ = … = f_p+2 = 0, 0_p+2 ≠ 0 for a fixed, but arbitrary p ≥ 0 such that p + 2 ≤ s (then f_p+2 is necessarily a homomorphism from (G, +) to (K, +).

Let l ∈ ℕ or l = ∞. We present a criterion for the extensibility of Φ_s to a homomorphism Φ_s+l from (G, +) to L_s+1 ¹ (L_∞ ¹, if l = ∞), by proving that such an extension (continuation) exists iff the component functions f_n of Φ_s with s - p ≤ n ≤ min(s - p + l - 1,s) are certain polynomials in f_P+2 (see Theorem 1). We also formulate the problem in the language of truncated formal power series in one indeterminate X over K. The somewhat easier situation f ₁ ≠ 1 will be studied in a separate paper.