Puiseux expansion

In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate T. They were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850. For example, the series

is a Puiseux series in T.

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.

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 Laurent series. This statement is also referred to as Puiseux's theorem, being an expression of the original Puiseux theorem in modern abstract language. Puiseux series are generalized by Hahn series.

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