Annales Mathematicae Silesianae 35 (2021), no. 1, 55–76 DOI: 10.2478/amsil-2020-0020 FAMILIES OF COMMUTING FORMAL POWER SERIES AND FORMAL FUNCTIONAL EQUATIONS Harald Fripertinger , Ludwig Reich Dedicated to Zygfryd Kominek on the occasion of his 75th birthday Abstract. In this paper we describe families of commuting invertible formal power series in one indeterminate over C, using the method of formal functional equations. We give a characterization of such families where the set of multipli- ers (first coefficients) σ of its members F (x) = σx + ::: is infinite, in particular of such families which are maximal with respect to inclusion, so called fami- lies of type I. The description of these families is based on Aczél–Jabotinsky differential equations, iteration groups, and on some results on normal forms of invertible series with respect to conjugation. 1. Introduction: Basic concepts and formulation of the main problem In this paper we want to describe families of commuting invertible formal power series in one indeterminate over C, using the method of formal func- tional equations. We give a characterization of such families where the set of Received: 20.05.2020. Accepted: 19.08.2020. Published online: 06.10.2020. (2020) Mathematics Subject Classification: 39B12, 39B52, 13F25, 30D05. Key words and phrases: commuting formal power series, maximal families of commut- ing formal power series, maximal abelian subgroups in Γ, formal functional equations, formal partial differential equations, Aczél–Jabotinsky equation, Briot–Bouquet equation, formal iteration groups of type I, ring of formal power series over C. c 2020 The Author(s). This is an Open Access article distributed under the terms of the Creative Commons Attribution License CC BY (http://creativecommons.org/licenses/by/4.0/). 56 Harald Fripertinger, Ludwig Reich multipliers (first coefficients) σ of its members F (x) = σx + ::: is infinite, in particular of such families which are maximal with respect to inclusion, so called families of type I. The description of these families is based on Aczél– Jabotinsky differential equations, iteration groups, and on some results on normal forms of invertible series with respect to conjugation. The study of families of commuting formal series started with [21]. The method of formal functional equations was so far applied to the con- struction of iteration groups of formal series and for solving the (associated) system of cocycle equations (see [3, 4, 5, 6, 7]). The basic idea of this method is to replace the coefficient functions by certain polynomials in one ore more variables. The situation of maximal families of commuting power series where the set of multipliers is finite, is technically more complicate and will be considered in a separate paper, also using the method of formal functional equations. Now we proceed by giving basic notations and results, in some places with indications of proofs, for the convenience of the reader. Let C[[x]] be the ring of formal power series in one indeterminate x over C. For a detailed introduction to formal power series we refer the reader to [1] and [8]. Together with addition + and multiplication · the set C[[x]] forms P n a commutative ring. If F 6= 0, then the order of F (x) = n≥0 cnx 2 C[[x]], cn 2 C, is defined as ord(F ) = minfn ≥ 0 j cn 6= 0g. Moreover, ord(0) = 1. According to the order topology in C[[x]], the series F; G 2 C[[x]] are “close” if P n ord(F − G) is “large”. For F; G 2 C[[x]], F (x) = n≥0 cnx , ord(G) ≥ 1, the composition ◦ of formal series is defined as n X ν (F ◦ G)(x) = F (G(x)) = lim cν [G(x)] n!1 ν=0 which exists in the order topology. Substitution of a fixed G 2 C[[x]] yields a homomorphism C[[x]] ! C[[x]]. The composition of formal series is associative, F ◦ (G ◦ H) = (F ◦ G) ◦ H, F; G; H 2 C[[x]], ord(G); ord(H) ≥ 1. With Γ = (Γ; ◦) we denote the group of invertible formal power series with respect to composition, thus Γ = fF 2 C[[x]] j ord(F ) = 1g = fF 2 C[[x]] j F (x) = c1x + : : : ; c1 6= 0g: Let Γ1 be the subgroup of Γ with c1 = 1, 2 Γ1 = fF 2 C[[x]] j F (x) = x + c2x + :::g: The group Γ is not abelian, e.g., for F (x) = −x and G(x) = x + x2 we have (F ◦ G)(x) = −x − x2 and (G ◦ F )(x) = −x + x2, thus F ◦ G 6= G ◦ F . Families of commuting formal power series and formal functional equations 57 Let C[[x; y]] := (C[[x]])[[y]] be the ring of formal power series in two in- determinates over C, then C[[x; y]] = C[[y; x]]. An element of C[[x; y]] can be written as X ν µ XX ν µ F (x; y) = cνµx y = cνµx y : ν,µ≥0 µ≥0 ν≥0 We will also need the ring C[[x; y; z]] in three indeterminates, and use the fact that (C[y])[[x]] < C[[x; y]]. P n Differentiation is a formal process in C[[x]]. Let F (x) = n≥0 cnx , then d X X F (x) = F 0(x) = nc xn−1 = (n + 1)c xn: dx n n+1 n≥1 n≥0 Differentiation satisfies the chain rule (F ◦ G)0(x) = F 0(G(x)) · G0(x). Considering a family (F (t; x))t2C of formal power series X n F (t; x) = cn(t)x ; t 2 C; n≥0 where all the coefficient functions cn are holomorphic in a neighborhood U(0) of 0, we define @ X F (t; x) = c0 (t)xn; t 2 U(0): @t n n≥0 In this situation the mixed chain rule @ @ F (G(t; x)) = F 0(G(t; x)) · G(t; x); t 2 U(0); @t @t holds true. We also consider formal partial differentiation in C[[x; y]]. Each F (x; y) 2 C[[x; y]] can be written as X n F (x; y) = Pn(y)x 2 (C[[y]])[[x]] n≥0 and @ X F (x; y) = P 0 (y)xn: @y n n≥0 58 Harald Fripertinger, Ludwig Reich Two formal power series F; G 2 Γ are conjugate if there exists some S 2 Γ such that G = S−1 ◦ F ◦ S: Let E be the group of complex roots of 1. It is a subgroup of the multiplicative group C∗. 2 Remark 1. Let F (x) = ρx + c2x + ::: 2 Γ, ρ 62 E. Then there exists a unique series S 2 Γ1 such that −1 (S ◦ F ◦ S)(x) = ρx =: Lρ(x): We say that F can be linearized, or F is linearizable. Proof. We determine all solutions S 2 Γ1 of Schröder’s equation (S) F (S(x)) = S(ρx): The left hand side can be expanded as X n F (S(x)) = ρS(x) + cn[S(x)] n≥2 2 3 = ρx + (ρs2 + c2)x + (ρs3 + 2c2s2ρ + c3)x + ::: X n = ρx + (ρsn + pn(ρ, s2; : : : ; sn−1; c2; : : : ; cn))x n≥2 with polynomials pn(t; u2;: : : ;un−1; v2; : : : ; vn) 2 C[t; u2; : : : ; un−1; v2; : : : ; vn], P n n n ≥ 2. The right hand side equals ρx+ n≥2 snρ x . Since ρ is not a complex root of unity, ρn 6= ρ for all n ≥ 2, and comparison of coefficients yields: 2 (ρ − ρ)s2 = c2, whence s2 is uniquely determined. In general for n ≥ 2 we obtain inductively n (ρ − ρ)sn = pn(ρ, s2; : : : ; sn−1; c2; : : : ; cn) and sn is uniquely determined. Remark 2. Assume that ρ 62 E, F^(x) = σx+::: 2 Γ, and Lρ ◦F^ = F^ ◦Lρ. Then F^(x) = σx. If F (x) = ρx + :::, F (x) = S(ρS−1(x)) for some S 2 Γ, F^(x) = σx + ::: 2 Γ, and F ◦ F^ = F^ ◦ F , then F^(x) = S(σS−1(x)). Families of commuting formal power series and formal functional equations 59 Proof. The first assertion is obtained by direct calculation. F commutes −1 −1 with F^ if and only if S ◦ Lρ ◦ S ◦ F^ = F^ ◦ S ◦ Lρ ◦ S which is equivalent −1 −1 to Lρ ◦ S ◦ F^ ◦ S = S ◦ F^ ◦ S ◦ Lρ and the assertion follows from the first part. Consider some σ = e2πi`=m 2 E, gcd(`; m) = 1, m ≥ 1. Then X nm+1 N(x) = σx + dnm+1x n≥1 is a semicanonical form with respect to σ. Remark 3. Let σ 2 E be primitive of order m>1, and F (x) = σx+:::2Γ. There exists some S 2 Γ1 such that −1 X nm+1 (S ◦ F ◦ S)(x) = σx + dnm+1x ; n≥1 i.e. a semicanonical form with respect to σ. In general, S is not uniquely determined. (Cf. [20, Section 2].) Proof. When σ is a root of unity primitive of order m > 1, then in general (S) cannot be satisfied by choosing coefficients in a suitable way, since in the comparison of coefficients of xn for n ≡ 1 mod m the term σn − σ vanishes, which means that pn(σ; s2; : : : ; sn−1; c2; : : : ; cn) must vanish, and sn is not determined by this equation. Therefore, we study the equation X nm+1 F (S(x)) = S(σx + dnm+1x ): n≥1 P n Its left hand side is σx + n≥2(σsn + pn(σ; s2; : : : ; sn−1; c2; : : : ; cn))x as in the proof of Remark 1. The right hand side can be expanded as X n n σx + (σ sn + qn(σ; s2; : : : ; sn−1; dm+1; : : : ; db(n−1)=mcm+1))x n≥2 with polynomials qn(t; u2; : : : ; un−1; v1; : : : ; vb(n−1)=mc) 2 C[t; u2; : : : ; un−1; v1; : : : ; vb(n−1)=mc], n ≥ 2.