- Puiseux expansion
In
mathematics , a Puiseux expansion is aformal power series expansion of analgebraic function . Puiseux's theorem is a classicalexistence theorem for such an expansion, in the case of one variable.If "K" is an
algebraically closed field of characteristic 0, thealgebraic closure of thefield of fractions of the ring:"K"
"T"of formal power series in the
indeterminate "T" can be described as the union of the formalLaurent 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 number s, these Puiseux series have non-zeroradius of convergence , and so provideanalytic function s 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} = 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 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 KT .The name is for
Victor Puiseux (1820-1883). The theory was at least implicit in the original use of theNewton 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]
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