Puiseux expansion

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In mathematics, a Puiseux expansion is a formal power series expansion of an algebraic function. Puiseux's theorem is a classical existence theorem for such an expansion, in the case of one variable.

If K is an algebraically closed field of characteristic 0, the algebraic closure of the field of fractions of the ring

K[[T]]

of formal power series in the indeterminate T can be described as the union of the formal Laurent series fields in all the fractional powers

T^1/n

for integers n ≥ 1 (this is not true if char(K) = p > 0). This means that locally near a point P an algebraic curve can be parametrised by a power series in some fixed T^1/n. In the interesting case when P is a singular point, there may be more than one branch. The (several) formal power series that result are called the Puiseux expansion(s), relative to P.

When the field K is the complex numbers, these Puiseux series have non-zero radius of convergence, and so provide analytic functions in terms of a fractional-power variable.

We can also define the field of transfinite Puiseux series as follows. Take K to be any field.

Define

$K[[T^\mathbb{Q}]] = \{f(T)=\sum_{\alpha\in\mathbb{Q}}a_\alpha t^\alpha,a_\alpha\in K, \mathrm{\;s.t.\;} \text{Supp}(f) = \{\alpha | a_\alpha\neq 0\}\mathrm{\; is\; well\; ordered}\}.$

One can show that if K is an algebraically closed field (e.g. $\mathbb{C}, \overline{\mathbb{F}}_q$ ), then $K[[T^\mathbb{Q}]]$ is also an algebraically closed field, and in general it is strictly bigger than $\overline{K((T))}$ , the algebraic closure of the field of fractions of $K [[T]]$ .