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, 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 Laurent series fields in all the fractional powers

: T 1/ n

for integers n ≥ 1. This means that locally at a point P on an algebraic curve can be parametrised by power series in some fixed T 1/ n . At a singular point, the interesting case, 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.

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]