[Paper Review] An Inversion Formula for Multivariate Power Series

Ural Bekbaev|arXiv (Cornell University)|Mar 17, 2012

Mathematical functions and polynomials1 references3 citations

TL;DR

This paper presents a novel inversion formula for formal multivariate power series $φ(x) = x + \text{higher-order terms}$, expressing the formal inverse as $\varphi^{-1}(x) = x + \sum_{m=1}^{\infty} \sum_{k=0}^{m} (-1)^k \binom{m}{k} \varphi^{\circ k}(x)$, where $\varphi^{\circ k}$ denotes $k$-fold composition. The derivation relies on a symmetric product of matrices and generating functions in the ring of formal power series, establishing a non-recursive, combinatorial inversion formula valid over fields of characteristic zero.

ABSTRACT

For formal multivariate power series $φ(x)$ an inversion formula of the form $$ φ^{-1}(x)=x +\sum_{m=1}^{\infty}\sum_{k=0}^m (-1)^k(m k)φ^{\circ k}(x) is offered$$.

Motivation & Objective

• To derive a closed-form inversion formula for formal multivariate power series $\varphi(x) = x + \text{higher-order terms}$.
• To establish a connection between formal inversion and iterated composition via a novel symmetric matrix product operation.
• To provide a non-recursive, combinatorial expression for the inverse series using binomial coefficients and iterated compositions.
• To investigate conditions under which the inverse of a polynomial map remains polynomial, via convergence and vanishing of series tails.

Proposed method

• Introduces a symmetric product $\odot$ on block matrices indexed by multi-indices, generalizing matrix multiplication to formal power series.
• Defines the matrix representation $M_{\varphi}$ of a power series $\varphi$, encoding its homogeneous components via multi-indexed blocks.
• Uses the exponential-like map $e^{\odot A}$ to represent the composition of power series in terms of matrix exponentiation under $\odot$.
• Applies the identity $M_{\psi \circ \varphi} = e^{\odot M_{\varphi}} M_{\psi}$ to relate composition to matrix operations.
• Derives the inverse via the Neumann series expansion of $(e^{\odot M_{\varphi}})^{-1}$, leading to the binomial-type sum over iterated compositions.
• Establishes a recurrence for coefficients of the inverse via block matrix equations in the $Mat(F)$ algebra.

Experimental results

Research questions

• RQ1 Can a closed-form inversion formula be derived for formal multivariate power series using only composition and binomial coefficients?
• RQ2 Under what conditions does the infinite series $\sum_{m=1}^{\infty} \sum_{k=0}^{m} (-1)^k \binom{m}{k} \varphi^{\circ k}(x)$ terminate or vanish?
• RQ3 When is the formal inverse of a polynomial map also a polynomial map, and how can this be detected algebraically?
• RQ4 How does the symmetric matrix product $\odot$ enable the construction of a non-associative-like but associative algebraic structure for formal power series?

Key findings

• The inverse of a formal power series $\varphi(x) = x + \text{higher-order terms}$ is given explicitly by $\varphi^{-1}(x) = x + \sum_{m=1}^{\infty} \sum_{k=0}^{m} (-1)^k \binom{m}{k} \varphi^{\circ k}(x)$.
• The formula is derived using the symmetric product $\odot$ and the matrix exponential $e^{\odot M_{\varphi}}$, with $M_{\varphi}$ encoding the series' coefficients.
• The inverse series exists and is unique in the ring of formal power series over any field of characteristic zero.
• If $\sum_{m=m_0}^{\infty} \sum_{k=0}^{m} (-1)^k \binom{m}{k} \varphi^{\circ k}(x) = 0$ for some $m_0$, then $\varphi^{-1}(x)$ is a polynomial map.
• The recurrence $N^m_1 = -\sum_{k=1}^{m-1} \left( \sum_{|\alpha|=k, \|\alpha\|=m} \frac{(M^1_1)^{\odot \alpha_1} \odot \cdots}{\alpha!} \right) N^k_1$ computes the coefficients of the inverse series.
• The set of polynomial maps with polynomial inverses forms a group under composition, and the formula provides a criterion for membership in this group.