Faà di Bruno's formula and inversion of power series

Samuel Johnston; Joscha Prochno

doi:10.1016/j.aim.2021.108080

Faà di Bruno's formula and inversion of power series

Samuel Johnston, Joscha Prochno

Department of Mathematical Sciences

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Abstract

Faà di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesised version of Faà di Bruno's formula in higher dimensions, providing a combinatorial expression for the derivatives of chain compositions of functions in terms of sums over labelled trees. We give several applications of this formula, including a new involution formula for the inversion of multivariate power series. We use this framework to outline a combinatorial approach to studying the invertibility of polynomial mappings, giving a purely combinatorial restatement of the Jacobian conjecture. Our methods extend naturally to the non-commutative case, where we prove a free version of Faà di Bruno's formula for multivariate power series in free indeterminates, and use this formula as a tool for obtaining a new inversion formula for free power series.

Original language	English
Article number	108080
Journal	Advances in Mathematics
Volume	395
Early online date	19 Nov 2021
DOIs	https://doi.org/10.1016/j.aim.2021.108080
Publication status	Published - 24 Feb 2022

Funding

The authors are extremely grateful to an anonymous referee whose suggestions have greatly improved this article. We would also like to thank Michael Anshelevich for directing us towards several useful references. SJ and JP have been supported by the Austrian Science Fund (FWF) Project P32405 “Asymptotic Geometric Analysis and Applications” of which JP is principal investigator. JP has also been supported by a Visiting International Professor Fellowship from the Ruhr University Bochum and its Research School PLUS. Finally, we gratefully acknowledge the support of the Oberwolfach Research Institute for Mathematics, where several discussion were held during the workshop “New Perspectives and Computational Challenges in High Dimensions” (Workshop ID 2006b). SJ and JP have been supported by the Austrian Science Fund (FWF) Project P32405 “Asymptotic Geometric Analysis and Applications” of which JP is principal investigator. JP has also been supported by a Visiting International Professor Fellowship from the Ruhr University Bochum and its Research School PLUS.

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10.1016/j.aim.2021.108080

1911.07458.PDFAccepted author manuscript, 482 KBLicence: CC BY-NC-ND

https://arxiv.org/abs/1911.07458Licence: CC BY-NC-ND

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Faà di Bruno's formula and inversion of power series

Abstract

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