Puiseux series

In mathematics, Puiseux series are a generalization of power series, first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850, that allow for negative and fractional exponents of the indeterminate T. A Puiseux series in the indeterminate T is a Laurent series in T^1/n, where n is a positive integer. A Puiseux series may be written as:

where ${\displaystyle k_{0}}$ is an integer and ${\displaystyle n}$ is a positive integer.

Puiseux's theorem, sometimes also called Newton–Puiseux theorem, asserts that, given a polynomial equation ${\displaystyle P(x,y)=0}$, its solutions in y, viewed as functions of x, may be expanded as Puiseux series that are convergent in some neighbourhood of the origin (0 excluded, in the case of a solution that tends to infinity at the origin). In other words, every branch of an algebraic curve may be locally (in terms of x) described by a Puiseux series.

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