Some Algebraic Aspects of the Center Problem for Ordinary Differential Equations

Brudnyi, Alexander

doi:10.1007/s12346-010-0018-5

Cited by 7 publications

(7 citation statements)

References 17 publications

(29 reference statements)

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“…with a i ∈ R for all i ∈ N (this inclusion depends on the choice of the coordinate z). The main goal of this note is the following result, that answers a question raised by E. Ghys (see [8], §3.3, or also [5], Problem 4.15).…”

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confidence: 79%

Surface Groups In The Group Of Germs Of Analyticdiffeomorphisms In One Variable

Cantat¹,

Cerveau²,

Guirardel³

et al. 2019

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confidence: 79%

Surface Groups In The Group Of Germs Of Analyticdiffeomorphisms In One Variable

Cantat¹,

Cerveau²,

Guirardel³

et al. 2019

Preprint

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“…4.7]) the group G[[r]] contains two-generator discrete subgroups which are neither abelian nor free (see also [NY] for further results in this direction). In turn, in [Br, Problem 4.15] we asked with regard to the center problem for families of Abel differential equations whether the fundamental groups of orientable compact Riemann surfaces are embeddable to G [[r]]. In this paper we answer this question affirmatively.…”

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confidence: 84%

“…Since all non-exceptional fundamental groups of compact Riemann surfaces (i.e., distinct from the fundamental groups of non-orientable surfaces of Euler characteristic 1, 0 or −1) are fully residually free (see [B1]), they are embeddable to G [[r]]. This answers [Br,Problem 4.15].…”

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confidence: 91%

Subgroups of the Group of Formal Power Series with the Big Powers Condition

Brudnyi

2019

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We study the structure of discrete subgroups of the group G[[r]] of complex formal power series under the operation of composition of series. In particular, we prove that every finitely generated fully residually free group is embeddable to G [[r]]. Main ResultLet G[[r]] be the prounipotent group of formal power series of the form r + ∞ i=1 c i r i+1 , c i ∈ C, i ∈ N, under the operation • of composition of series. In the paper we study the problem on the structure of discrete subgroups of G [[r]]. The problem is of importance, in particular, in connection with the classification of local analytic foliations and the holonomy of local differential equations (see, e.g.,[NY] and references therein). The deep results of [EV] show that in contrast to free prounipotent groups (see [LM, Cor. 4.7]) the group G[[r]] contains two-generator discrete subgroups which are neither abelian nor free (see also [NY] for further results in this direction). In turn, in [Br, Problem 4.15] we asked with regard to the center problem for families of Abel differential equations whether the fundamental groups of orientable compact Riemann surfaces are embeddable to G [[r]]. In this paper we answer this question affirmatively. Our approach is purely group-theoretical and can be applied to a wide class of prounipotent groups.To formulate the main result of the paper we introduce several definitions. Let G be a group and u = (u 1 , . . . , u k ), k ∈ N, be a tuple of non-trivial elements of G. We say that u is commutation-free1 for any integers α 1 , . . . , α k ≥ n.Definition 1.1. Group G satisfies the big powers condition if every commutation-free tuple in G is independent.

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“…Namely, it was shown in [8] that in the case where F (x, y), G(x, y) are homogeneous and of the same degree, the Poincare problem reduces to the center problem for Abel equation (1). The center problem for the Abel equation and its modifications are the subject of many recent papers involving different approaches and techniques (see e. g. [1], [4], [5], [6], [7], [9], [10], [11], [12] and the bibliography therein). Set…”

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confidence: 99%