Multivariate formal power series

Summary

A formal power series in $X_1, \dots, X_n$ over a commutative ring $R$ is a possibly infinite sum $\sum a_\alpha X^\alpha$ over all monomials $X^\alpha$. No convergence condition is imposed. These series form the ring $R[\![X_1, \dots, X_n]\!]$, which extends the one-variable formal power series ring and contains the polynomial ring $R[X_1, \dots, X_n]$. Over a field, this ring is the basic example of a complete local ring and of a regular local ring of dimension $n$.