TITLE: What's the difference of naming a polynomial ring as $\mathbb{C}\{ x,y\}$ and $\mathbb{C} [x,y]$?
QUESTION [5 upvotes]: 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?

REPLY [7 votes]: $\mathbb{C}\{x,y\}$ 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 $x,y$.