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.

 

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