Puiseux series

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 the Newton–Puiseux theorem, asserts that, given a polynomial equation formula_2, its solutions in , viewed as functions of , 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 ) 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 formal 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.
Formal definition.
If "K" is a field (such as the complex numbers) then we can define the field of Puiseux series with coefficients in "K" informally as the set of expressions of the form
where formula_4 is a positive integer and formula_5 is an arbitrary integer. In other words, Puiseux series differ from Laurent series in that they allow for fractional exponents of the indeterminate, as long as these fractional exponents have bounded denominator (here "n"). Just as with Laurent series, Puiseux series allow for negative exponents of the indeterminate as long as these negative exponents are bounded below (here by formula_5). 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 and then performing the operation in the corresponding field of formal Laurent series of formula_9.
In other words, the field of Puiseux series with coefficients in "K" is the union of the fields formula_10 (where "n" ranges over the positive integers), where each element of the union is a field of formal Laurent series over formula_11 (considered as an indeterminate), and where each such field is considered as a subfield of the ones with larger "n" by rewriting the fractional exponents to use a larger denominator (so, for example, formula_12 is identified with formula_13 ).
This yields a formal definition of the field of Puiseux series: it is the direct limit of the direct system, indexed over the non-zero natural numbers "n" ordered by divisibility, whose objects are all formula_14 (the field of formal Laurent series, which we rewrite as formula_15 for clarity), with a morphism formula_16 being given, whenever "m" divides "n", by formula_17.
Valuation and order.
The Puiseux series over a field "K" form a valued field with value group formula_18 (the rationals): the "valuation" formula_19 of a series
as above is defined to be the smallest rational formula_21 such that the coefficient formula_22 of the term with exponent formula_21 is non-zero (with the usual convention that the valuation of 0 is +∞). The coefficient formula_22 in question is typically called the "valuation coefficient" of "f".
This valuation in turn defines a (translation-invariant) distance (which is ultrametric), hence a topology on the field of Puiseux series by letting the distance from "f" to 0 be formula_25. This justifies "a posteriori" the notation
as the series in question does, indeed, converge to "f" in the Puiseux series field (this is in contrast to Hahn series which "cannot" be viewed as converging series).
If the base field "K" is ordered, then the field of Puiseux series over "K" is also naturally (“lexicographically”) ordered as follows: a non-zero Puiseux series "f" with 0 is declared positive whenever its valuation coefficient is so. Essentially, this means that any positive rational power of the indeterminate "T" is made positive, but smaller than any positive element in the base field "K".
If the base field "K" is endowed with a valuation "w", then we can construct a different valuation on the field of Puiseux series over "K" by letting the valuation formula_27 be formula_28 where formula_29 is the previously defined valuation (formula_22 is the first non-zero coefficient) and ω is infinitely large (in other words, the value group of formula_31 is formula_32 ordered lexicographically, where Γ is the value group of "w"). Essentially, this means that the previously defined valuation "v" is corrected by an infinitesimal amount to take into account the valuation "w" given on the base field.
Algebraic closedness of Puiseux series.
One essential property of Puiseux series is expressed by the following theorem, attributed to Puiseux (for formula_33) but which was implicit in Newton's use of the Newton polygon as early as 1671 and therefore known either as Puiseux's theorem or as the Newton–Puiseux theorem:
Theorem: If "K" is an algebraically closed field of characteristic zero, then the field of Puiseux series over "K" is the algebraic closure of the field of formal Laurent series over "K".
Very roughly, the proof proceeds essentially by inspecting the Newton polygon of the equation and extracting the coefficients one by one using a valuative form of Newton's method. Provided algebraic equations can be solved algorithmically in the base field "K", then the coefficients of the Puiseux series solutions can be computed to any given order.
For example, the equation formula_34 has solutions
and
(one readily checks on the first few terms that the sum and product of these two series are 1 and formula_37 respectively; this is valid whenever the base field "K" has characteristic different from 2).
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. The example of the Artin–Schreier equation formula_38 shows this: reasoning with valuations shows that "X" should have valuation formula_39, and if we rewrite it as formula_40 then
and one shows similarly that formula_42 should have valuation formula_43, and proceeding in that way one obtains the series
since this series makes no sense as a Puiseux series—because the exponents have unbounded denominators—the original equation has no solution. However, such Eisenstein equations are essentially the only ones not to have a solution, because, if "K" is algebraically closed of characteristic "p">0, then the field of Puiseux series over "K" is the perfect closure of the maximal tamely ramified extension of formula_14.
Similarly to the case of algebraic closure, there is an analogous theorem for real closure: if "K" is a real closed field, then the field of Puiseux series over "K" is the real closure of the field of formal Laurent series over "K". (This implies the former theorem since any algebraically closed field of characteristic zero is the unique quadratic extension of some real-closed field.)
There is also an analogous result for p-adic closure: if "K" is a "p"-adically closed field with respect to a valuation "w", then the field of Puiseux series over "K" is also "p"-adically closed.
Puiseux expansion of algebraic curves and functions.
Algebraic curves.
Let "X" be an algebraic curve given by an affine equation formula_46 over an algebraically closed field "K" of characteristic zero, and consider a point "p" on "X" which we can assume to be (0,0). We also assume that "X" is not the coordinate axis "x" = 0. Then a "Puiseux expansion" of (the "y" coordinate of) "X" at "p" is a Puiseux series "f" having positive valuation such that formula_47.
More precisely, let us define the "branches" of "X" at "p" to be the points "q" of the normalization "Y" of "X" which map to "p". For each such "q", there is a local coordinate "t" of "Y" at "q" (which is a smooth point) such that the coordinates "x" and "y" can be expressed as formal power series of "t", say formula_48 (since "K" is algebraically closed, we can assume the valuation coefficient to be 1) and formula_49: then there is a unique Puiseux series of the form formula_50 (a power series in formula_11), such that formula_52 (the latter expression is meaningful since formula_53 is a well-defined power series in "t"). This is a Puiseux expansion of "X" at "p" which is said to be associated to the branch given by "q" (or simply, the Puiseux expansion of that branch of "X"), and each Puiseux expansion of "X" at "p" is given in this manner for a unique branch of "X" at "p".
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 formula_54 (whose normalization is a line with coordinate "t" and map formula_55) has two branches at the double point (0,0), corresponding to the points "t" = +1 and "t" = −1 on the normalization, whose Puiseux expansions are formula_56 and formula_57 respectively (here, both are power series because the "x" coordinate is étale at the corresponding points in the normalization). At the smooth point (−1,0) (which is "t" = 0 in the normalization), it has a single branch, given by the Puiseux expansion formula_58 (the "x" coordinate ramifies at this point, so it is not a power series).
The curve formula_59 (whose normalization is again a line with coordinate "t" and map formula_60), on the other hand, has a single branch at the cusp point (0,0), whose Puiseux expansion is formula_61.
Analytic convergence.
When formula_33 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 "n"-th root of "x", they converge for small enough formula_63, hence define an analytic parametrization of each branch of "X" in the neighborhood of "p" (more precisely, the parametrization is by the "n"-th root of "x").
Generalizations.
Levi-Civita field.
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 formula_64 where the support of the coefficients (that is, the set of "e" such that formula_65) 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 "A" for any given bound "A". For example, formula_66 is not a Puiseux series, but it is the limit of a Cauchy sequence of Puiseux series; in particular, it is the limit of formula_67 as formula_68. 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.
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 formula_18 or formula_70). 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.