Puiseux series

In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series

${\displaystyle {\begin{aligned}x^{-2}&+2x^{-1/2}+x^{1/3}+2x^{11/6}+x^{8/3}+x^{5}+\cdots \\&=x^{-12/6}+2x^{-3/6}+x^{2/6}+2x^{11/6}+x^{16/6}+x^{30/6}+\cdots \end{aligned}}}$

is a Puiseux series in the indeterminate x. Puiseux series were first introduced by Isaac Newton in 1676^[1] and rediscovered by Victor Puiseux in 1850.^[2]

The definition of a Puiseux series includes that the denominators of the exponents must be bounded. So, by reducing exponents to a common denominator n, a Puiseux series becomes a Laurent series in an nth root of the indeterminate. For example, the example above is a Laurent series in ${\displaystyle x^{1/6}.}$ Because a complex number has n nth roots, a convergent Puiseux series typically defines n functions in a neighborhood of 0.

Puiseux's theorem, sometimes also called the Newton–Puiseux theorem, asserts that, given a polynomial equation ${\displaystyle P(x,y)=0}$ with complex coefficients, its solutions in y, viewed as functions of x, may be expanded as Puiseux series in x that are convergent in some neighbourhood of 0. In other words, every branch of an algebraic curve may be locally described by a Puiseux series in x (or in x − x₀ when considering branches above a neighborhood of x₀ ≠ 0).

Using modern terminology, Puiseux's theorem asserts that the set of Puiseux series over an algebraically closed field of characteristic 0 is itself an algebraically closed field, called the field of Puiseux series. It is the algebraic closure of the field of formal Laurent series, which itself is the field of fractions of the ring of formal power series.