# Puiseux series

In this note, we discuss about Puiseux series and its appearance when we solve the equation $f(x,y) = 0$. We refer to the book “Algebraic curves” of R.Walker.

Puiseux series are fractional power series:

$\bar{a}(x)= a_1x^{\frac{m_1}{n_1}}+a_2x^{\frac{m_2}{n_2}}+ \dots$ where $a_i \ne 0, m_1/n_1 < m_2/n_2 < \dots$.

Order of series: $O(\bar{a}(x)) = m_1/n_1$.

Theorem. $K(x)^*$ is algebraically closed.

($K(x)^*$ – the fieldof fractional power series).

By the proof of this theorem, we can see that $f(x,y) = 0$ (an algebraic curve), we can solve $\bar{y}(x)$ (Puiseux series) such that $f(x, \bar{y}) = 0$.