It's a short question, namely
When can one think about the henselianization $A^h$ of a local ring $A$ as the "algebraic part" of its completion $\hat{A}$?
It seems to be true for $k[\vec{x}]$, where $\vec{x}$ is a bunch of indeterminants.
And clearly there is a morphism from $A^h$ to $\hat{A}$.
Are there more general situation where $A^h$ embeds into $\hat{A}$ as the algebraic part?
(In case this is not a good question, let me ask a different one: what's a good way to think about $A^h$?)
