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I sometimes see both notations and I am led (maybe misled) to believe that they are the same thing. What is the formal difference between both of them? Or there isn't any?

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$\mathbb{C}\{x,y\}$ is the ring of convergent power series in $x,y$; $\mathbb{C}[x,y]$ is the ring of polynomials in $x,y$. The third related object you often see is $\mathbb{C}[[x,y]]$; this is the ring of formal power series in $x,y$.

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  • $\begingroup$ This makes things cleares than I hoped :) I was having this doubt, since the material I'm reading said $dim_{\mathbb{C}}\frac{\mathcal{O}}{a\mathcal{O}+b\mathcal{O}}$ is finite under some conditions, where $a,b\in\mathcal{O}=\mathbb{C}\{ x,y\}$, $\mathcal{O}$ the ring of function germs at a singularity. $\endgroup$ – Marra May 7 '13 at 12:59
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    $\begingroup$ I'm glad it's helpful! Yes, you could also call $\mathbb{C}\{x,y\}$ the ring of (analytic) function germs at the origin. $\endgroup$ – Rhys May 7 '13 at 13:42

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