A note on multiplicative roots of multivariable formal power series

Suppose that we are given a formal power series of many variables with coefficients in R R (or C C ) and we wish to compute its n -th (multiplicative) root. As expected the coefficients of the root must satisfy a system of infinitely many equations. We present such a system of equations that strictly corresponds with the system for n -th root of a formal power series of one variable. Through an illustrative example, we demonstrate that the case of formal power series of many variables differs significantly from the one-variable case with respect to the existence of roots.