C}\ll x\gg$ of $f(x,y)=0$ with $\ord(Y)>0$ there corresponds a unique place of $\cal C$ with center at the origin. Conversely, to each place $(X(t),Y(t))$ of $\cal C$ with center at the origin there correspond $\ord(X)$ roots of $f(x,y)=0$, each of order greater than zero.

If $Y(x)$ is a Puiseux series solving $f(x,y)=0$, $\ord(Y)>0$, and $n$ is the least integer for which $Y(x) \in {\Bbb C}((x^{1\over
n}))$ (i.e., $\nu(Y)=n$), then we set $x^{1\over n}=t$, and $(t^n, Y(t^n))$ is a local parametrization with center at the origin. The solutions of $f(x,y)$ of order 0 correspond to places with center on the $y$-axis but different from the origin, and the solutions of negative order correspond to places at infinity (places with center at an infinity point).

Note that several different Puiseux series may correspond to equivalent local parametrizations, and then these series provide a unique place. More precisely, let $Y(x)=\sum_{i\geq r}a_ix^{i/n}$ be a Puiseux series with ramification index $\nu(Y)=n$. The series $\sigma_{\epsilon}(Y)$, $\epsilon^n=1$, are called the [*conjugates*]{} of $Y$, where $$\sigma_{\epsilon}(Y)=\sum_{i\geq r}\epsilon^ia_ix^{i/n}.$$ The set of all (distinct) conjugates of $Y$ is called the [*conjugacy class*]{} of $Y$. The number of different conjugates of $Y$ is $\nu(Y)$. Two Puiseux series provide the same place 