A note on multiplicative roots of multivariable formal power series

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.