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Let t be the uniformizing parameter of the local ring $O_{C,P}$, where $C \subseteq A^2_k$ is an irreducible variety, k a field, and $P \in C$. Let $\alpha \in O_{C,P}$. Prove that there exist uniquely defined $a_0,...,a_n \in k$ and $\alpha \in O_{C,P}$ such that $\alpha = a_0+a_1t+...+a_nt^n+ \alpha_nt^{n+1}$ for every $n \in \mathbb{N}$.

What I am having a hard time with is piecing together the local ring with the field. This is what I know so far:

The local ring is a DVR, so $t$ irreducible, such that for all $\alpha \in O_{C,P}$, there exists a unique unit in $O_{C,P}$ and unique $n \in \mathbb{N}$ such that $\alpha = ut^m$. Clearly the $a_0,...,a_n$ are invertible in k, but I don't know what I can say about them in $O_{C,P}$. Can I say that $a_0+a_1t+...+a_nt^n$ is a unit in $O_{C,P}$?

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    $\begingroup$ Apply repeatly the fact that $O_{C,P}=k+tO_{C,P}$. $\endgroup$
    – user143488
    May 10, 2014 at 22:03
  • $\begingroup$ I would get $\alpha= a_0+t(a_1+t(a_2+t(....t(a_n+ \alpha_nt))..)$ . I don't see how that helps. Wouldn't I want to separate the t's and then everything but the $\alpha_n$ is a unit in $O_{C,P}$? $\endgroup$
    – math1234567
    May 11, 2014 at 7:06
  • $\begingroup$ The coefficients $a_i$ are not necessarily units. Look at the example $\alpha=t$. $\endgroup$
    – user143488
    May 11, 2014 at 9:22
  • $\begingroup$ I think I am getting the names mixed up. I meant that the polynomial with the $a_i$'s is an invertible element of the local ring. $\endgroup$
    – math1234567
    May 11, 2014 at 10:05
  • $\begingroup$ Isn't the same example a counterexample or I misunderstand something ? $\endgroup$
    – user143488
    May 11, 2014 at 21:59

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