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{{Short description|Generalization of power series to rational powers}}
{{Technical|reason=The article should start by providing examples, definitions, and theorems in the setting of undergraduate-level complex analysis, which will make it useful to a fairly wide audience (engineers, physicists, mathematics undergraduates, etc.). Only after that should it provide the formulation informed by modern algebra (involving concepts such as "functor," "union of fields," "direct limit," etc.), which is useful only to a much narrower audience.|date=April 2011}}

In [[mathematics]], '''Puiseux series''' are a generalization of [[power series]] that allow for negative and fractional exponents of the [[Indeterminate (variable)|indeterminate]] ''T''. They were first introduced by [[Isaac Newton]] in 1676<ref>{{harvtxt|Newton|1960}}</ref> and rediscovered by [[Victor Puiseux]] in 1850.<ref name="Puiseux1850">Puiseux (1850, 1851)</ref> For example, the series
:<math> T^{-2} + 2T^{-1/2} + T^{1/3} + 2T^{11/6} + T^{8/3} + 2T^{25/6} + T^5 + 2T^{13/2} + \cdots</math>
is a Puiseux series in&nbsp;''T''.

'''Puiseux's theorem''', sometimes also called the '''Newton–Puiseux theorem''', asserts that, given a [[polynomial equation]] <math>P(x,y)=0</math>, its solutions in {{mvar|y}}, viewed as functions of {{mvar|x}}, may be expanded as Puiseux series that are [[convergent series|convergent]] in some [[neighbourhood (mathematics)|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 {{mvar|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 [[Laurent series|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]].

== Formal definition ==
If ''K'' is a [[field (mathematics)|field]] (such as the [[complex number]]s) then we can define the field of Puiseux series with coefficients in ''K'' informally as the set of expressions of the form
:<math>f = \sum_{k=k_0}^{+\infty} c_k T^{k/n}</math>
where <math>n</math> is a positive integer and <math>k_0</math> 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 <math>k_0</math>). Addition and multiplication are as expected: for example, 
:<math> (T^{-1} + 2T^{-1/2} + T^{1/3} + \cdots) + (T^{-5/4} - T^{-1/2} + 2 + \cdots) = T^{-5/4} + T^{-1} + T^{-1/2} + 2 + \cdots</math>
and
:<math> (T^{-1} + 2T^{-1/2} + T^{1/3} + \cdots) \cdot (T^{-5/4} - T^{-1/2} + 2 + \cdots) = T^{-9/4} + 2T^{-7/4} - T^{-3/2} + T^{-11/12} + 4T^{-1/2} + \cdots</math>.
One might define them by first "upgrading" the denominator of the exponents to some common denominator {{math|''N''}} and then performing the operation in the corresponding field of formal Laurent series of <math>T^{1/N}</math>.

In other words, the field of Puiseux series with coefficients in ''K'' is the union of the fields <math>K(\!(T^{1/n})\!)</math> (where ''n'' ranges over the positive integers), where each element of the union is a field of formal Laurent series over <math>T^{1/n}</math> (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, <math>T^{1/2}</math> is identified with <math>T^{15/30}</math> ).{{clarify|This is problematic as written: it's only when one denominator divides the other that we get inclusion, right?|date=April 2017}}

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 [[Divisor|divisibility]], whose objects are all <math>K(\!(T)\!)</math> (the field of formal Laurent series, which we rewrite as <math>K(\!(T_n)\!)</math> for clarity), with a morphism <math>K(\!(T_m)\!) \to K(\!(T_n)\!)</math> being given, whenever ''m'' divides ''n'', by <math>T_m \mapsto (T_n)^{n/m}</math>.

=== Valuation and order ===
The Puiseux series over a field ''K'' form a [[Valuation (algebra)|valued]] field with value group <math>\Q</math> (the [[rationals]]): the ''valuation'' <math>v(f)</math> of a series

:<math>f = \sum_{k=k_0}^{+\infty} c_k T^{k/n}</math>

as above is defined to be the smallest rational <math>k/n</math> such that the coefficient <math>c_k</math> of the term with exponent <math>k/n</math> is non-zero (with the usual convention that the valuation of 0 is +∞). The coefficient <math>c_k</math> in question is typically called the ''valuation coefficient'' of&nbsp;''f''.

This valuation in turn defines a (translation-invariant) [[Metric (mathematics)|distance]] (which is [[Ultrametric space|ultrametric]]), hence a [[Topological space|topology]] on the field of Puiseux series by letting the distance from ''f'' to 0 be <math>\exp(-v(f))</math>. This justifies ''a posteriori'' the notation

:<math>f = \sum_{k=k_0}^{+\infty} c_k T^{k/n}</math>

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 field|ordered]], then the field of Puiseux series over ''K'' is also naturally (“[[Lexicographical order|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 <math>\hat w(f)</math> be <math>\omega\cdot v + w(c_k),</math> where <math>v=k/n</math> is the previously defined valuation (<math>c_k</math> is the first non-zero coefficient) and ω is infinitely large (in other words, the value group of <math>\hat w</math> is <math>\Q \times \Gamma</math> 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<ref name="Puiseux1850"/> (for <math>K=\C</math>) but which was implicit in [[Isaac Newton|Newton]]'s use of the [[Newton polygon]] as early as 1671<ref>Newton (1736)</ref> and therefore known either as Puiseux's theorem or as the Newton–Puiseux theorem:<ref name="Kedlaya2001Intro">cf. Kedlaya (2001), introduction</ref>

'''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''.<ref>cf. Eisenbud (1995), corollary&nbsp;13.15 (p.&nbsp;295)</ref>

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 <math>X^2 - X = T^{-1}</math> has solutions

:<math>X = T^{-1/2} + \frac{1}{2} + \frac{1}{8}T^{1/2} - \frac{1}{128}T^{3/2} + \cdots</math>

and

:<math>X = -T^{-1/2} + \frac{1}{2} - \frac{1}{8}T^{1/2} + \frac{1}{128}T^{3/2} + \cdots</math>

(one readily checks on the first few terms that the sum and product of these two series are 1 and <math>-T^{-1}</math> respectively; this is valid whenever the base field ''K'' has characteristic different from&nbsp;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 theory|Artin–Schreier]] equation <math>X^p - X = T^{-1}</math> shows this: reasoning with valuations shows that ''X'' should have valuation <math>-\frac{1}{p}</math>, and if we rewrite it as <math>X = T^{-1/p} + X_1</math> then

:<math>X^p = T^{-1} + {X_1}^p,\text{ so }{X_1}^p - X_1 = T^{-1/p}</math>

and one shows similarly that <math>X_1</math> should have valuation <math>-\frac{1}{p^2}</math>, and proceeding in that way one obtains the series

:<math>T^{-1/p} + T^{-1/p^2} + T^{-1/p^3} + \cdots;</math>

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's criterion|Eisenstein equations]] are essentially the only ones not to have a solution, because, if ''K'' is algebraically closed of characteristic ''p''&gt;0, then the field of Puiseux series over ''K'' is the perfect closure of the maximal tamely [[Ramification (mathematics)|ramified]] extension of <math>K(\!(T)\!)</math>.<ref name="Kedlaya2001Intro" />

Similarly to the case of algebraic closure, there is an analogous theorem for [[Real closed field|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''.<ref>Basu &amp;al (2006), chapter&nbsp;2 ("Real Closed Fields"), theorem&nbsp;2.91 (p.&nbsp;75)</ref> (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-adically closed field|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.<ref>Cherlin (1976), chapter&nbsp;2 ("The Ax–Kochen–Ershof Transfer Principle"), §7 ("Puiseux series fields")</ref><!-- I don't know whether it is *the* p-adic closure or whether it can be larger: can someone settle this? -->

== Puiseux expansion of algebraic curves and functions ==

=== Algebraic curves ===
Let ''X'' be an [[algebraic curve]]<ref>We assume that ''X'' is [[Algebraic variety|irreducible]] or, at least, that it is reduced and that it does not contain the ''y'' coordinate axis.</ref> given by an affine equation <math>F(x,y)=0</math> over an algebraically closed field ''K'' of characteristic zero, and consider a point ''p'' on ''X'' which we can assume to be&nbsp;(0,0). We also assume that ''X'' is not the coordinate axis ''x''&nbsp;=&nbsp;0. Then a ''Puiseux expansion'' of (the ''y'' coordinate of) ''X'' at ''p'' is a Puiseux series ''f'' having positive valuation such that <math>F(x,f(x))=0</math>.

More precisely, let us define the ''branches'' of ''X'' at ''p'' to be the points ''q'' of the [[Noether normalization lemma|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 <math>x = t^n + \cdots</math> (since ''K'' is algebraically closed, we can assume the valuation coefficient to be&nbsp;1) and <math>y = c t^k + \cdots</math>: then there is a unique Puiseux series of the form <math>f = c T^{k/n} + \cdots</math> (a power series in <math>T^{1/n}</math>), such that <math>y(t)=f(x(t))</math> (the latter expression is meaningful since <math>x(t)^{1/n} = t+\cdots</math> 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''.<ref>Shafarevich (1994), II.5, pp.&nbsp;133–135</ref><ref>Cutkosky (2004), chapter&nbsp;2, pp.&nbsp;3–11</ref>

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.<ref>Puiseux (1850), p.&nbsp;397</ref>

For example, the curve <math>y^2 = x^3 + x^2</math> (whose normalization is a line with coordinate ''t'' and map <math>t \mapsto (t^2-1,t^3-t)</math>) has two branches at the double point (0,0), corresponding to the points ''t''&nbsp;=&nbsp;+1 and ''t''&nbsp;=&nbsp;−1 on the normalization, whose Puiseux expansions are <math>y = x + \frac{1}{2}x^2 - \frac{1}{8}x^3 + \cdots</math> and <math>y = - x - \frac{1}{2}x^2 + \frac{1}{8}x^3 + \cdots</math> respectively (here, both are power series because the ''x'' coordinate is [[Étale morphism|étale]] at the corresponding points in the normalization). At the smooth point (−1,0) (which is ''t''&nbsp;=&nbsp;0 in the normalization), it has a single branch, given by the Puiseux expansion <math>y = -(x+1)^{1/2} + (x+1)^{3/2}</math> (the ''x'' coordinate ramifies at this point, so it is not a power series).

The curve <math>y^2 = x^3</math> (whose normalization is again a line with coordinate ''t'' and map <math>t \mapsto (t^2,t^3)</math>), on the other hand, has a single branch at the [[Cusp (singularity)|cusp point]] (0,0), whose Puiseux expansion is <math>y = x^{3/2}</math>.

=== Analytic convergence ===
When <math>K=\C</math> is the field of complex numbers, the Puiseux expansion of an algebraic curve (as defined above) is [[Radius of convergence|convergent]] in the sense that for a given choice of ''n''-th root of ''x'', they converge for small enough <math>|x|</math>, 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 ===
{{main|Levi-Civita field}}
The field of Puiseux series is not [[Complete metric space|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 <math>f = \sum_e c_e T^e,</math> where the support of the coefficients (that is, the set of ''e'' such that <math>c_e \neq 0</math>) 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, <math>\sum_{k=1}^{+\infty} T^{k+\frac{1}{k}}</math> is not a Puiseux series, but it is the limit of a [[Cauchy sequence]] of Puiseux series; in particular, it is the limit of <math>\sum_{k=1}^{N} T^{k+\frac{1}{k}}</math> as <math>N \to +\infty</math>. 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,<ref>{{cite journal |last1=Poonen |first1=Bjorn |year=1993 |title=Maximally complete fields |journal=Enseign. Math. |volume=39 |pages=87–106 |ref=harv}}</ref><ref>{{cite journal |last1=Kaplansky |first1=Irving |year=1942 |title=Maximal Fields with Valuations |journal=Duke Math. J. |volume=9 |pages=303–321|ref=harv |doi=10.1215/s0012-7094-42-00922-0}}</ref> hence the opportunity of completing it even more.

=== Hahn series ===
{{main|Hahn series}}
[[Hahn series]] are a further (larger) generalization of Puiseux series, introduced by [[Hans Hahn (mathematician)|Hans Hahn]] in the course of the proof of his [[Hahn embedding theorem|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-order|well-ordered subset]] of the value group (usually <math>\Q</math> or <math>\R</math>). 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.<ref>Kedlaya (2001)</ref>

== Notes ==
<references />

==See also==
* [[Laurent series]]
* [[Madhava series]]
* [[Newton polynomial|Newton's divided difference interpolation]]
* [[Padé approximant]]

== References ==
* {{Cite book | last1=Basu | first1=Saugata | last2=Pollack | first2=Richard | last3=Roy | first3=Marie-Françoise | title=Algorithms in Real Algebraic Geometry | publisher=[[Springer-Verlag]] | series=Algorithms and Computations in Mathematics 10 | edition=2nd | year=2006 | doi=10.1007/3-540-33099-2 | isbn=978-3-540-33098-1 | url=http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted1.html }}
* {{Cite book | last=Cherlin | first=Greg | title=Model Theoretic Algebra Selected Topics | publisher=[[Springer-Verlag]] | series=Lecture Notes in Mathematics 521 | year=1976 | isbn=978-3-540-07696-4 | url=http://www.springerlink.com/content/x5l812m32426/ }}
* {{Cite book | last=Cutkosky | first=Steven Dale | title=Resolution of Singularities | publisher=American Mathematical Society | series=[[Graduate Studies in Mathematics]] 63 | year=2004 | isbn=0-8218-3555-6 }}
* {{Cite book | last=Eisenbud | first=David | authorlink=David Eisenbud | title=Commutative Algebra with a View Toward Algebraic Geometry | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics]] 150 | year=1995 | isbn=3-540-94269-6 }}
* {{Cite journal | doi=10.1090/S0002-9939-01-06001-4 | last=Kedlaya | first=Kiran Sridhara | year=2001 | title=The algebraic closure of the power series field in positive characteristic | journal=Proc. Amer. Math. Soc. | volume=129 | pages=3461–3470 | ref=harv | postscript=<!--None--> }}
* {{Citation|last=Newton|first=Isaac|author-link=Isaac Newton|translator-last =Colson|translator-first =John|translator-link = John Colson|title=The method of fluxions and infinite series; with its application to the geometry of curve-lines|place =London|publisher =Henry Woodfall|orig-year =1671
|year=1736|pages =378|language =|url =https://archive.org/details/methodoffluxions00newt|doi =|id =|isbn =|mr =|zbl =|jfm = }} (Translated from Latin)
* {{Cite book|first=Isaac|last=Newton|year=1960|chapter=letter to Oldenburg dated 1676 Oct 24|title=The correspondence of Isaac Newton|volume=II|publisher=Cambridge University press|pages=126–127|ref=harv|postscript=<!--None-->|isbn=0-521-08722-8|url-access=registration|url=https://archive.org/details/correspondence0007newt}}
* {{Cite journal | last=Puiseux | first=Victor Alexandre | author-link=Victor Puiseux | year=1850 | title=Recherches sur les fonctions algébriques | journal=J. Math. Pures Appl. | volume=15 | pages=365–480 | ref=harv | url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1850_1_15_A24_0.pdf }}
* {{Cite journal | last=Puiseux | first=Victor Alexandre | author-link=Victor Puiseux | year=1851 | title=Nouvelles recherches sur les fonctions algébriques | journal=J. Math. Pures Appl. | volume=16 | pages=228–240 | ref=harv | url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1851_1_16_A15_0.pdf }}
* {{Cite book | last=Shafarevich | first=Igor Rostislavovich | authorlink=Igor Shafarevich | title=Basic Algebraic Geometry | publisher=[[Springer-Verlag]] | year=1994 | edition=2nd | isbn=3-540-54812-2 }}

== External links ==
*{{springer|title=Branch point|id=p/b017500}}
*[http://mathworld.wolfram.com/PuiseuxSeries.html Puiseux series at MathWorld]
*[http://mathworld.wolfram.com/PuiseuxsTheorem.html Puiseux's Theorem at MathWorld]
*[http://planetmath.org/encyclopedia/FractionalPowerSeries.html Puiseux series at PlanetMath]

{{Isaac Newton}}

[[Category:Commutative algebra]]
[[Category:Algebraic curves]]
[[Category:Mathematical series]]