A note on multiplicative roots of multivariable formal power series

TL;DR

The paper investigates multiplicative $n$-th roots of multivariable formal power series over ${\\mathbb{R}}$ or ${\\mathbb{C}}$, revealing how the multivariable setting differs from the one-variable theory. It introduces a constructive, equation-based framework that mirrors the one-variable root equations, distinguishing unit and nonunit cases via the initial block and providing explicit recursions for coefficient computation. A key result is that, for units, the existence of an $n$-th root is equivalent to the initial block having an $n$-th root, with roots built by choosing $g_{ heta}$ so that $g_{ heta}^n=f_{ heta}$ and then determining higher coefficients through recurrences. For nonunits, the work reduces the problem to the initial monomial via a decomposition $f=(f_eta X^eta) g$ with $g_{ heta}=1$, leading to a system of linear and polynomial equations, and a matrix relation that may admit multiple solutions, illustrating the intricate nature of multivariable roots.

Abstract

Suppose that we are given a formal power series of many variables with coefficients in $\mathbb{R}$ (or $\mathbb{C}$) and we want to compute its $n$-th (multiplicative) root. As can be expected coefficients of the root have to satisfy a system of infinitely many equations. We present such a system of equations that strictly corresponds with the system for $n$-th of a formal power series of one variable. With help of an example we show that the case of formal power series of many variables is very different from the one variable case with respect to the existence of roots.

Theorems & Definitions (17)

• Definition 1
• Remark 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Proposition 7
• proof
• Remark 8
• Theorem 9