Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fa\`a di Bruno's formula and inversion of power series   Fa\`a 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 synthesized version of Fa\`a di Bruno's formula in higher dimensions,
providing a combinatorial expression for the derivatives of chain compositions
$F^{(1)} \circ \ldots \circ F^{(m)}$ of functions $F^{(l)} : \mathbb{R}^N \to
\mathbb{R}^N$ 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 Fa\`a 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.

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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: nlin
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The categories of this paper are: cond-mat
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
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The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th