Puiseux expansion

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

:$KT^mathbb\{Q\}=\; left\{left.f(T)=sum\_\{alphainmathbb\{Qa\_alpha\; t^alpha,a\_alphain\; K\; ~\; ight|~\; ext\{Supp\}(f)\; =\; \{alpha\; |\; a\_alpha\; eq\; 0\}mathrm\{;\; is;\; well;\; ordered\}\; ight\}.$

One can show that if "K" is an algebraically closed field (e.g. $mathbb\{C\},\; overline\{mathbb\{F\_q$), then $KT^mathbb\{Q\}$ 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 $KT$.

The name is for Victor Puiseux (1820-1883). The theory was at least implicit in the original use of the Newton polygon.

External links

* [http://mathworld.wolfram.com/PuiseuxSeries.html Puiseux series at MathWorld]
* [http://mathworld.wolfram.com/PuiseuxsTheorem.html Puiseux's Theorem at MathWorld]
* [http://planetmath.org/encyclopedia/FractionalPowerSeries.html Puiseux series at PlanetMath]