Puiseux series

Power series with rational exponents

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

is a Puiseux series in the indeterminate x. Puiseux series were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850. 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 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 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 when considering branches above a neighborhood of ). 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.

If the base field is ordered, then the field of Puiseux series over is also naturally (“lexicographically”) ordered as follows: a non zero Puiseux series with 0 is declared positive whenever its valuation coefficient is so. Essentially, this means that any positive rational power of the indeterminate is made positive, but smaller than any positive element in the base field
If the base field is endowed with a valuation , then we can construct a different valuation on the field of Puiseux series over by letting the valuation be where is the previously defined valuation ( is the first non zero coefficient) and is infinitely large (in other words, the value group of is ordered lexicographically, where is the value group of ). Essentially, this means that the previously defined valuation is corrected by an infinitesimal amount to take into account the valuation given on the base field.

As early as 1671, Isaac Newton implicitly used Puiseux series and proved the following theorem for approximating with series the roots of algebraic equations whose coefficients are functions that are themselves approximated with series or polynomials. For this purpose, he introduced the Newton polygon, which remains a fundamental tool in this context. Newton worked with truncated series, and it is only in 1850 that Victor Puiseux 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. Moreover, the proof provides an algorithm for computing these Puiseux series, and, when working over the complex numbers, the resulting series are convergent. In modern terminology, the theorem can be restated as: the field of Puiseux series over an algebraically closed field of characteristic zero, and the field of convergent Puiseux series over the complex numbers, are both algebraically closed.

The way of applying recursively the method of the Newton polygon has been described precedingly. As each application of the method may increase, in the ramified case, the denominators of exponents (valuations), it remains to prove that one reaches the regular case after a finite number of iterations (otherwise the denominators of the exponents of the resulting series would not be bounded, and this series would not be a Puiseux series. By the way, it will also be proved that one gets exactly as many Puiseux series solutions as expected, that is the degree of in y. Without loss of generality, one can suppose that that is, Indeed, each factor y of provides a solution that is the zero Puiseux series, and such factors can be factored out. As the characteristic is supposed to be zero, one can also suppose that is a square free polynomial, that is that the solutions of are all different. Indeed, the square free factorization uses only the operations of the field of coefficients for factoring into square free factors than can be solved separately. (The hypothesis of characteristic zero is needed, since, in characteristic p, the square free decomposition can provide irreducible factors, such as that have multiple roots over an algebraic extension.) 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 a polygon is the sum of the lengths of its edges. With the hypothesis the length of the Newton polygon of P is its degree in y, that is the number of its roots. The length of an edge of the Newton polygon is the number of roots of a given valuation. This number equals the degree of the previously defined polynomial

The ramified case corresponds thus to two (or more) solutions that have the same initial term(s). As these solutions must be distinct (square free hypothesis), they must be distinguished after a finite number of iterations. That is, one gets eventually a polynomial 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 numbersis an algebraically closed field that contains the univariate polynomial ring with complex coefficients.

Let be an algebraic curve given by an affine equation over an algebraically closed field of characteristic zero, and consider a point on which we can assume to be We also assume that is not the coordinate axis Then a Puiseux expansion of (the coordinate of) at is a Puiseux series having positive valuation such that
More precisely, let us define the branches of at to be the points of the normalization of which map to For each such , there is a local coordinate of at (which is a smooth point) such that the coordinates and can be expressed as formal power series of , say (since is algebraically closed, we can assume the valuation coefficient to be 1) and : then there is a unique Puiseux series of the form (a power series in ), such that (the latter expression is meaningful since is a well defined power series in ). This is a Puiseux expansion of at which is said to be associated to the branch given by (or simply, the Puiseux expansion of that branch of ), and each Puiseux expansion of at is given in this manner for a unique branch of at
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. For example, the curve (whose normalization is a line with coordinate and map ) has two branches at the double point (0,0), corresponding to the points and on the normalization, whose Puiseux expansions are and respectively (here, both are power series because the coordinate is étale at the corresponding points in the normalization). At the smooth point (which is in the normalization), it has a single branch, given by the Puiseux expansion (the coordinate ramifies at this point, so it is not a power series). The curve (whose normalization is again a line with coordinate and map ), on the other hand, has a single branch at the cusp point , whose Puiseux expansion is .

When 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 th root of , they converge for small enough , hence define an analytic parametrization of each branch of in the neighborhood of (more precisely, the parametrization is by the th root of ).

The field of Puiseux series is not complete as a metric space. Its completion, called the Levi Civita field, can be described as follows: it is the field of formal expressions of the form where the support of the coefficients (that is, the set of e such that ) is the range of an increasing sequence of rational numbers that either is finite or tends to In other words, such series admit exponents of unbounded denominators, provided there are finitely many terms of exponent less than for any given bound For example, is not a Puiseux series, but it is the limit of a Cauchy sequence of Puiseux series; in particular, it is the limit of as However, even this completion is still not "maximally complete" in the sense that it admits non trivial extensions which are valued fields having the same value group and residue field, hence the opportunity of completing it even more.

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. In a Hahn series, instead of requiring the exponents to have bounded denominator they are required to form a well ordered subset of the value group (usually or ). 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). Using Hahn series, it is possible to give a description of the algebraic closure of the field of power series in positive characteristic which is somewhat analogous to the field of Puiseux series.