A note on multiplicative roots of multivariable formal power series

TL;DR

This paper investigates the existence and computation of multiplicative roots of multivariable formal power series, revealing significant differences from the single-variable case through a new system of equations and illustrative examples.

Contribution

It introduces a system of equations for multivariable roots that aligns with the one-variable case and highlights the distinct behavior in multiple variables.

Findings

Coefficients of roots satisfy an infinite system of equations.

Existence of roots in multivariable series differs markedly from the one-variable case.

The paper provides an example illustrating these differences.

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.