Puiseux expansion
From Wikipedia, the free encyclopedia
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
- T1/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 T1/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.\;} Supp(f) = \{\alpha | a_\alpha\neq 0\}\mathrm{\; is\; well\; ordered}\}.](../../../math/d/e/4/de4ac517c2c0df8ccfa36431d4442115.png)
One can show that if K is an algebraically closed field (e.g. 
), then ![K[[T^\mathbb{Q}]]](../../../math/0/2/3/0237eec6939e75aa712c09d5991bd88c.png)
is also an algebraically closed field, and in general it is strictly bigger than 
, the algebraic closure of the field of fractions of K[[T]].
The name is for Victor Puiseux (1820-1883). The theory was at least implicit in the original use of the Newton polygon.
