Puiseux series

Truncated Puiseux expansions for the cubic curve

y^{2}=x^{3}+x^{2}

at the double point

x=y=0

. Darker colors indicate more terms.

In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. They were first introduced by Isaac Newton in 1676^[1] and rediscovered by Victor Puiseux in 1850.^[2] For example, the series

{\begin{aligned}x^{-2}&+2x^{-1/2}+x^{1/3}+2x^{11/6}+x^{8/3}+x^{5}+\cdots \\&=x^{-12/6}+2x^{-3/6}+x^{2/6}+2x^{11/6}+x^{16/6}+x^{30/6}+\cdots \end{aligned}}

is a Puiseux series in $x$ .

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 a $n$ th root of the indeterminates. For example, the above series is a Laurent series in $x^{1/6}.$ As a number has $n$ $n$ th roots, a convergent Puiseux series defines, in general $n$ functions in a neighborhood of $0$ .

Puiseux's theorem, sometimes also called the Newton–Puiseux theorem, asserts that, given a polynomial equation $P(x,y)=0$ with complex coeffients, its solutions in $y$ , viewed as functions of $x$ , may be expanded as Puiseux series in $x$ that are convergent in some neighbourhood of the zero, (possibly zero excluded, in the case of a solution that tends to the infinity when $x$ tends to $0$ ). In other words, every branch of an algebraic curve may be locally described by a Puiseux series in $x$ (of in $x - x 0$ when considering branches above a neighborhood of $x 0 \neq 0$ ).

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.

Definition[edit]

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

f=\sum _{k=k_{0}}^{+\infty }c_{k}T^{k/n}

where $n$ is a positive integer and $k_{0}$ is an 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 $k_{0}$ ). Addition and multiplication are as expected: for example,

(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

and

(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 .

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 $T^{1/N}$ .

The Puiseux series with coefficients in $K$ form a field, which is the union

\bigcup _{n>0}K(\!(T^{1/n})\!)

of fields of formal Laurent series in $T^{1/n}$ (considered as an indeterminate).

This yields an alternative definition of the field of Puiseux series in terms of a direct limit. For every positive integer $n$ , let $T_{n}$ be an indeterminate (meant to represent ${\textstyle T^{1/n}}$ ), and $K(\!(T_{n})\!)$ be the field of formal Laurent series in $T_{n}.$ If $m$ divides $n$ , the mapping $T_{m}\mapsto (T_{n})^{n/m}$ induces a field homomorphism $K(\!(T_{m})\!)\to K(\!(T_{n})\!),$ and these homomorphisms form a direct system that has the field of Puiseux series as 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).

Valuation[edit]

A nonzero Puiseux series $f$ can be uniquely written as

f=\sum _{k=k_{0}}^{+\infty }c_{k}T^{k/n}

with $c_{k_{0}}\neq 0.$ The valuation

v(f)={\frac {k_{0}}{n}}

of $f$ is the smallest exponent for the natural order of the rational numbers, and the corresponding coefficient ${\textstyle c_{k_{0}}}$ is called the initial coefficient or valuation coefficient of f. The valuation of the zero series is $+\infty .$

The function $v$ is a valuation and makes the Puiseux series a valued field, with the additive group $\mathbb {Q}$ of the rational numbers as its valuation group.

As for every valued fields, the valuation defines a ultrametric distance by the formula $d(f,g)=\exp(-v(f-g)).$ For this distance, the field of Puiseux series is a complete metric space. The notation

f=\sum _{k=k_{0}}^{+\infty }c_{k}T^{k/n}

expresses that a Puiseux is the limit of its partial sums.

Convergent Puiseux series[edit]

Puiseux series provided by Newton–Puiseux theorem are convergent in the sense that there is a nonempty neighborhood of zero in which they are convergent (0 excluded if the valuation is negative). More precisely, let

f=\sum _{k=k_{0}}^{+\infty }c_{k}T^{k/n}

be a Puiseux series with complex coefficients. There is a real number $r$ , called the radius of convergence such that the series converges if $T$ is substitued for a nonzero complex number $t$ of absolute value less than $r$ , and $r$ is the largest number with this property. A Puiseux series is convergent if it has a nonzero radius of convergence.

Because a nonzero complex number has $n$ $n$ th roots, some care must be taken for the substitution: a specific $n$ th root of $t$ , say $x$ , must be chosen. Then the substitution consists of replacing $T^{k/n}$ by $x^{k}$ for every $k$ .

The existence of the radius of convergence results from the similar existence for a power series, applied to ${\textstyle T^{-k_{0}/n}f,}$ considered as a power series in $T^{1/n}.$

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).

Valuation and order on coefficients[edit]

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 ${\hat {w}}(f)$ be $\omega \cdot v+w(c_{k}),$ where $v=k/n$ is the previously defined valuation ( $c_{k}$ is the first non-zero coefficient) and ω is infinitely large (in other words, the value group of ${\hat {w}}$ is $\mathbb {Q} \times \Gamma$ 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.

Newton–Puiseux theorem[edit]

As early as 1671,^[3] Isaac Newton used implicitly 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^[2] introduced the concept of (non-truncated) Puiseux series and proved the theorem that is now known as Puiseux's theorem or Newton–Puiseux theorem.^[4] The theorem asserts that, given an algebraic equation whose coefficients are polynomials or, more generally, Puiseux series over a field of characteristic zero, then each 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 a field of characteristic zero, and the field of convergent Puiseux series over the complex numbers are both algebraically closed.

Newton polygon[edit]

The theorem[edit]

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.^[5]

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 $X^{2}-X=T^{-1}$ has solutions

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

and

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

(one readily checks on the first few terms that the sum and product of these two series are 1 and $-T^{-1}$ 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 $X^{p}-X=T^{-1}$ shows this: reasoning with valuations shows that X should have valuation $-{\frac {1}{p}}$ , and if we rewrite it as $X=T^{-1/p}+X_{1}$ then

X^{p}=T^{-1}+{X_{1}}^{p},{\text{ so }}{X_{1}}^{p}-X_{1}=T^{-1/p}

and one shows similarly that $X_{1}$ should have valuation $-{\frac {1}{p^{2}}}$ , and proceeding in that way one obtains the series

T^{-1/p}+T^{-1/p^{2}}+T^{-1/p^{3}}+\cdots ;

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 $K(\!(T)\!)$ .^[4]

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.^[6] (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.^[7]

Puiseux expansion of algebraic curves and functions[edit]

Algebraic curves[edit]

Let X be an algebraic curve^[8] given by an affine equation $F(x,y)=0$ 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 $F(x,f(x))=0$ .

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 $x=t^{n}+\cdots$ (since K is algebraically closed, we can assume the valuation coefficient to be 1) and $y=ct^{k}+\cdots$ : then there is a unique Puiseux series of the form $f=cT^{k/n}+\cdots$ (a power series in $T^{1/n}$ ), such that $y(t)=f(x(t))$ (the latter expression is meaningful since $x(t)^{1/n}=t+\cdots$ 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.^[9]^[10]

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.^[11]

For example, the curve $y^{2}=x^{3}+x^{2}$ (whose normalization is a line with coordinate t and map $t\mapsto (t^{2}-1,t^{3}-t)$ ) 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 $y=x+{\frac {1}{2}}x^{2}-{\frac {1}{8}}x^{3}+\cdots$ and $y=-x-{\frac {1}{2}}x^{2}+{\frac {1}{8}}x^{3}+\cdots$ 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 $y=-(x+1)^{1/2}+(x+1)^{3/2}$ (the x coordinate ramifies at this point, so it is not a power series).

The curve $y^{2}=x^{3}$ (whose normalization is again a line with coordinate t and map $t\mapsto (t^{2},t^{3})$ ), on the other hand, has a single branch at the cusp point (0,0), whose Puiseux expansion is $y=x^{3/2}$ .

Analytic convergence[edit]

When $K=\mathbb {C}$ 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 $|x|$ , 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[edit]

Levi-Civita field[edit]

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 $f=\sum _{e}c_{e}T^{e},$ where the support of the coefficients (that is, the set of e such that $c_{e}\neq 0$ ) 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, $\sum _{k=1}^{+\infty }T^{k+{\frac {1}{k}}}$ is not a Puiseux series, but it is the limit of a Cauchy sequence of Puiseux series; in particular, it is the limit of $\sum _{k=1}^{N}T^{k+{\frac {1}{k}}}$ as $N\to +\infty$ . 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,^[12]^[13] hence the opportunity of completing it even more.

Hahn series[edit]

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 $\mathbb {Q}$ or $\mathbb {R}$ ). 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.^[14]

Notes[edit]

^ Newton (1960)
^ ^a ^b Puiseux (1850, 1851)
^ Newton (1736)
^ ^a ^b cf. Kedlaya (2001), introduction
^ cf. Eisenbud (1995), corollary 13.15 (p. 295)
^ Basu &al (2006), chapter 2 ("Real Closed Fields"), theorem 2.91 (p. 75)
^ Cherlin (1976), chapter 2 ("The Ax–Kochen–Ershof Transfer Principle"), §7 ("Puiseux series fields")
^ We assume that X is irreducible or, at least, that it is reduced and that it does not contain the y coordinate axis.
^ Shafarevich (1994), II.5, pp. 133–135
^ Cutkosky (2004), chapter 2, pp. 3–11
^ Puiseux (1850), p. 397
^ Poonen, Bjorn (1993). "Maximally complete fields". Enseign. Math. 39: 87–106.
^ Kaplansky, Irving (1942). "Maximal Fields with Valuations". Duke Math. J. 9 (2): 303–321. doi:10.1215/s0012-7094-42-00922-0.
^ Kedlaya (2001)

References[edit]

Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise (2006). Algorithms in Real Algebraic Geometry. Algorithms and Computations in Mathematics 10 (2nd ed.). Springer-Verlag. doi:10.1007/3-540-33099-2. ISBN 978-3-540-33098-1.
Cherlin, Greg (1976). Model Theoretic Algebra Selected Topics. Lecture Notes in Mathematics 521. Springer-Verlag. ISBN 978-3-540-07696-4.^{[dead link]}
Cutkosky, Steven Dale (2004). Resolution of Singularities. Graduate Studies in Mathematics 63. American Mathematical Society. ISBN 0-8218-3555-6.
Eisenbud, David (1995). Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics 150. Springer-Verlag. ISBN 3-540-94269-6.
Kedlaya, Kiran Sridhara (2001). "The algebraic closure of the power series field in positive characteristic". Proc. Amer. Math. Soc. 129 (12): 3461–3470. doi:10.1090/S0002-9939-01-06001-4.
Newton, Isaac (1736) [1671], The method of fluxions and infinite series; with its application to the geometry of curve-lines, translated by Colson, John, London: Henry Woodfall, p. 378 (Translated from Latin)
Newton, Isaac (1960). "letter to Oldenburg dated 1676 Oct 24". The correspondence of Isaac Newton. II. Cambridge University press. pp. 126–127. ISBN 0-521-08722-8.
Puiseux, Victor Alexandre (1850). "Recherches sur les fonctions algébriques" (PDF). J. Math. Pures Appl. 15: 365–480.
Puiseux, Victor Alexandre (1851). "Nouvelles recherches sur les fonctions algébriques" (PDF). J. Math. Pures Appl. 16: 228–240.
Shafarevich, Igor Rostislavovich (1994). Basic Algebraic Geometry (2nd ed.). Springer-Verlag. ISBN 3-540-54812-2.
Walker, R.J. (1978). Algebraic Curves (PDF) (Reprint ed.). Springer-Verlag. ISBN 0-387-90361-5.

External links[edit]

"Branch point", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Puiseux series at MathWorld
Puiseux's Theorem at MathWorld
Puiseux series at PlanetMath

[1] Newton (1960)

[Puiseux1850-2] Puiseux (1850, 1851)

[3] Newton (1736)

[Kedlaya2001Intro-4] . Kedlaya (2001), introduction

[5] . Eisenbud (1995), corollary 13.15 (p. 295)

[6] Basu &al (2006), chapter 2 ("Real Closed Fields"), theorem 2.91 (p. 75)

[7] Cherlin (1976), chapter 2 ("The Ax–Kochen–Ershof Transfer Principle"), §7 ("Puiseux series fields")

[8] We assume that X is irreducible or, at least, that it is reduced and that it does not contain the y coordinate axis.

[9] Shafarevich (1994), II.5, pp. 133–135

[10] Cutkosky (2004), chapter 2, pp. 3–11

[11] Puiseux (1850), p. 397

[12] Poonen, Bjorn (1993). "Maximally complete fields". Enseign. Math. 39: 87–106.

[13] Kaplansky, Irving (1942). "Maximal Fields with Valuations". Duke Math. J. 9 (2): 303–321. doi:10.1215/s0012-7094-42-00922-0.

[14] Kedlaya (2001)

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

v t e Sir Isaac Newton
Publications	Fluxions (1671) De Motu (1684) Principia (1687; writing) Opticks (1704) Queries (1704) Arithmetica (1707) De Analysi (1711)
Other writings	Quaestiones (1661–1665) "standing on the shoulders of giants" (1675) Notes on the Jewish Temple (c. 1680) "General Scholium" (1713; "hypotheses non fingo" ) Ancient Kingdoms Amended (1728) Corruptions of Scripture (1754)
Contributions	Calculus fluxion Impact depth Inertia Newton disc Newton polygon Newton–Okounkov body Newton's reflector Newtonian telescope Newton scale Newton's metal Newton's cradle Spectrum Structural coloration
Newtonianism	Bucket argument Newton's inequalities Newton's law of cooling Newton's law of universal gravitation post-Newtonian expansion parameterized gravitational constant Newton–Cartan theory Schrödinger–Newton equation Newton's laws of motion Kepler's laws Newtonian dynamics Newton's method in optimization Apollonius's problem truncated Newton method Gauss–Newton algorithm Newton's rings Newton's theorem about ovals Newton–Pepys problem Newtonian potential Newtonian fluid Classical mechanics Corpuscular theory of light Leibniz–Newton calculus controversy Newton's notation Rotating spheres Newton's cannonball Newton–Cotes formulas Newton's method generalized Gauss–Newton method Newton fractal Newton's identities Newton polynomial Newton's theorem of revolving orbits Newton–Euler equations Newton number kissing number problem Newton's quotient Parallelogram of force Newton–Puiseux theorem Absolute space and time Luminiferous aether Newtonian series table
Personal life	Woolsthorpe Manor (birthplace) Cranbury Park (home) Early life Later life Religious views Occult studies Scientific Revolution Copernican Revolution
Relations	Catherine Barton (niece) John Conduitt (nephew-in-law) Isaac Barrow (professor) William Clarke (mentor) Benjamin Pulleyn (tutor) John Keill (disciple) William Stukeley (friend) William Jones (friend) Abraham de Moivre (friend)
Depictions	Newton by Blake (monotype) Newton by Paolozzi (sculpture)
Namesake	Newton (unit) Isaac Newton Institute Isaac Newton Medal Isaac Newton Telescope Isaac Newton Group of Telescopes Sir Isaac Newton Sixth Form Statal Institute of Higher Education Isaac Newton Newton International Fellowship
Categories	Isaac Newton

Puiseux series

Contents