# on functions with a single pole

Let $X$ a curve over an algebraically closed field $k$. $x$ a closed point. Let $F_{x}$ the completion at x of the function field of $X$. e chose an uniformizer $t$ such that $F_{x}=k((t))$.

Does it exists a integer $n\geq 0$ such that

$t^{-n}k[t^{-1}]\subset k[X-x]$

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No. $t$, even assuming it lives in the function field of $X$, could have a zero anywhere else on $X$. Indeed, for a curve of genus $g>0$, it must have a zero somewhere else. Then no power of $t^{-1}$ is in $k[X-x]$. –  Will Sawin Feb 7 at 4:59