Faà di Bruno's formula and inversion of power series

Johnston, Samuel G. G.; Prochno, Joscha

doi:10.1016/j.aim.2021.108080

Cited by 9 publications

(14 citation statements)

References 29 publications

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“…While the Jacobian problem most commonly lies in the remit of algebraic geometry, over the years several authors [1,17,20,27,29] have noted that the problem may be attacked using combinatorial arguments. We shall discuss this literature in further detail in Section 1.3, but let us give here a brief outline of what we believe to be the simplest formulation of this very concrete approach, which rests on the following three observations:…”

Section: Combinatorial Approach To the Jacobian Conjecturementioning

confidence: 99%

Random planar trees and the Jacobian conjecture

Bisi¹,

Dyszewski²,

Gantert³

et al. 2023

Preprint

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We develop a probabilistic approach to the celebrated Jacobian conjecture. This more than 80 year old problem from algebraic geometry states that any Keller map (i.e. any polynomial mapping F : C n → C n whose Jacobian determinant is a nonzero constant) has a compositional inverse which is also a polynomial. The Jacobian conjecture may be formulated in terms of a problem involving labellings of rooted trees; we give a new probabilistic derivation of this formulation using branching processes. Thereafter, we develop a simple and novel approach to the Jacobian conjecture in terms of a conjecture about shuffling subtrees of planar rooted d-ary trees. We show that such a shuffling conjecture is true in a certain asymptotic sense. We next use this machinery to prove an approximate version of the Jacobian conjecture, stating that inverses of Keller maps have small power series coefficients for their high degree terms.

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Section: Combinatorial Approach To the Jacobian Conjecturementioning

confidence: 99%

Random planar trees and the Jacobian conjecture

Bisi¹,

Dyszewski²,

Gantert³

et al. 2023

Preprint

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“…At the end of this section we give a recursive description of the inverse of an invertible series, which is simpler than Lagrange inversion formula [11,15]. Theorem 9.…”

Section: Formal Transformation Groupsmentioning

confidence: 99%

“…The unit is given by the series id = (x 1 , • • • , x n ). The inverse of multivariable series is given by the Lagrange inversion formula, a family of integer polynomials that is independent of R. There are many proofs of it, including Raney [21], Labelle [16], Gebelle [11], Haiman-Schmitt [12] and Johnston-Prochno [15]. In addition, see [10] for non-commutative case.…”

Section: Introductionmentioning

confidence: 99%

Periodic multivariate formal power series

Zhang¹

2022

Preprint

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A system of multivariate formal power series ϕ with a homogeneous decomposition ϕ = ∞ k=0 ϕ k is invertible under composition if ϕ 0 = 0 and det(ϕ 1 ) = 0. All invertible series over a field K form a formal transformation group G ∞ (n, K). We prove that every periodic series ϕ ∈ G ∞ (n, K) with ϕ 1 diagonalizable is conjugate to ϕ 1 . This classifies all periodic series in G ∞ (n, C). A constraint for a periodic series is obtained when its first term is a multiple of identity.

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“…The Faà di Bruno formula is a way to compute the higher order derivatives of the composition of two functions [27]. It is a useful tool to deal with power series [1,28] and can be seen both from the points of view of analysis [30] or of algebra [20].…”

Section: Taylor and Faà DI Bruno Formulaementioning

confidence: 99%

“…and define 𝐹 𝛿,𝑛 similarly to 𝐹 𝑛 p𝜃q in (28) with 𝑀 𝛿 instead of 𝑀 . The solution 𝜃 𝛿,𝑛 to 𝐹 𝛿,𝑛 p𝜃 𝛿,𝑛 q " 0 also solves…”

Section: 𝑛ñ8mentioning

confidence: 99%

Beyond the delta method

Lejay¹,

Mazzonetto²

2022

Preprint

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We give an asymptotic development of the maximum likelihood estimator (MLE), or any other estimator defined implicitly, in a way which involves the limiting behavior of the score and its higher-order derivatives. This development, which is explicitly computable, gives some insights about the non-asymptotic behavior of the renormalized MLE and its departure from its limit. We highlight that the results hold whenever the score and its derivative converge, including to non Gaussian limits.

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Faà di Bruno's formula and inversion of power series

Cited by 9 publications

References 29 publications

Random planar trees and the Jacobian conjecture

Random planar trees and the Jacobian conjecture

Periodic multivariate formal power series

Beyond the delta method

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