Let $R=k[X_1,...,X_n]$ be a polynomial ring over the field $k$ and let $\mathfrak m$ be a maximal ideal of $R$.
Question: Is the $\mathfrak m$-adic completion of $R$ a local ring ?
If $\mathfrak m=(X_1-a_1,\ldots,X_n-a_n)$ with $a_i \in k$ then the $\mathfrak m$-adic completion of $R$ is the local ring $k[[X_1-a_1,\ldots,X_n-a_n]]=k[[X_1,\ldots,X_n]]$. In particular, the question has an affirmative answer if $k$ is algebraically closed.
A related question is When is the completion of a ring a local ring ?.