Puiseux series

[2] The definition of a Puiseux series includes that the denominators of the exponents must be bounded.

Because a complex number has n nth roots, a convergent Puiseux series typically defines n functions in a neighborhood of 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.

If K is a field (such as the complex numbers), a Puiseux series with coefficients in K is an expression of the form where

Addition and multiplication are as expected: for example, and One might define them by first "upgrading" the denominator of the exponents to some common denominator N and then performing the operation in the corresponding field of formal Laurent series of

This yields an alternative definition of the field of Puiseux series in terms of a direct limit.

The fact that every field homomorphism is injective shows that this direct limit can be identified with the above union, and that the two definitions are equivalent (up to an isomorphism).

is the smallest exponent for the natural order of the rational numbers, and the corresponding coefficient

The function v is a valuation and makes the Puiseux series a valued field, with the additive group

As for every valued fields, the valuation defines a ultrametric distance by the formula

It is a part of Newton–Puiseux theorem that the provided Puiseux series have a positive radius of convergence, and thus define a (multivalued) analytic function in some neighborhood of zero (zero itself possibly excluded).

For this purpose, he introduced the Newton polygon, which remains a fundamental tool in this context.

[4] The theorem asserts that, given an algebraic equation whose coefficients are polynomials or, more generally, Puiseux series over a field of characteristic zero, every solution of the equation can be expressed as a Puiseux series.

For computing the Puiseux series that are roots of P (that is solutions of the functional equation

In summary, the valuation of a root of P must be the opposite of a slope of an edge of the Newton polynomial.

In summary, the Newton polynomial allows an easy computation of all possible initial terms of Puiseux series that are solutions of

One may apply to it the method of the Newton polygon, and iterate for getting the terms of the Puiseux series, one after the other.

and showing that one get a Puiseux series, that is, that the denominators of the exponents of x remain bounded.

The derivation with respect to y does not change the valuation in x of the coefficients; that is, and the equality occurs if and only if

In this context, one defines the length of an edge of a Newton polygon as the difference of the abscissas of its end points.

The length of an edge of the Newton polygon is the number of roots of a given valuation.

As these solutions must be distinct (square-free hypothesis), they must be distinguished after a finite number of iterations.

that is square free, and the computation can continue as in the regular case for each root of

As the iteration of the regular case does not increase the denominators of the exponents, This shows that the method provides all solutions as Puiseux series, that is, that the field of Puiseux series over the complex numbers is an algebraically closed field that contains the univariate polynomial ring with complex coefficients.

has solutions and (one readily checks on the first few terms that the sum and product of these two series are 1 and

As the powers of 2 in the denominators of the coefficients of the previous example might lead one to believe, the statement of the theorem is not true in positive characteristic.

[8][9] This existence of a formal parametrization of the branches of an algebraic curve or function is also referred to as Puiseux's theorem: it has arguably the same mathematical content as the fact that the field of Puiseux series is algebraically closed and is a historically more accurate description of the original author's statement.

is the field of complex numbers, the Puiseux expansion of an algebraic curve (as defined above) is convergent in the sense that for a given choice of

Hahn series are a further (larger) generalization of Puiseux series, introduced by Hans Hahn in the course of the proof of his embedding theorem in 1907 and then studied by him in his approach to Hilbert's seventeenth problem.

These were later further generalized by Anatoly Maltsev and Bernhard Neumann to a non-commutative setting (they are therefore sometimes known as Hahn–Mal'cev–Neumann series).