This article may be too technical for most readers to understand.(April 2011) (Learn how and when to remove this template message) |

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^{ [1] } and rediscovered by Victor Puiseux in 1850.^{ [2] } For example, the series

- Formal definition
- Valuation and order
- Algebraic closedness of Puiseux series
- Puiseux expansion of algebraic curves and functions
- Algebraic curves
- Analytic convergence
- Generalizations
- Levi-Civita field
- Hahn series
- Notes
- See also
- References
- External links

is a Puiseux series in *T*.

**Puiseux's theorem**, sometimes also called the **Newton–Puiseux theorem**, asserts that, given a polynomial equation , its solutions in y, viewed as functions of x, 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 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 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.

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 is a positive integer and 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 ). 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 .

In other words, the field of Puiseux series with coefficients in *K* is the union of the fields (where *n* ranges over the positive integers), where each element of the union is a field of formal Laurent series over (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, is identified with ).^{[ clarification needed ]}

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 (the field of formal Laurent series, which we rewrite as for clarity), with a morphism being given, whenever *m* divides *n*, by .

The Puiseux series over a field *K* form a valued field with value group (the rationals): the *valuation* of a series

as above is defined to be the smallest rational such that the coefficient of the term with exponent is non-zero (with the usual convention that the valuation of 0 is +∞). The coefficient 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 . 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 be where is the previously defined valuation ( is the first non-zero coefficient) and ω is infinitely large (in other words, the value group of is 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.

One essential property of Puiseux series is expressed by the following theorem, attributed to Puiseux^{ [2] } (for ) but which was implicit in Newton's use of the Newton polygon as early as 1671^{ [3] } and therefore known either as Puiseux's theorem or as the Newton–Puiseux theorem:^{ [4] }

**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 has solutions

and

(one readily checks on the first few terms that the sum and product of these two series are 1 and 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 shows this: reasoning with valuations shows that *X* should have valuation , and if we rewrite it as then

and one shows similarly that should have valuation , 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 .^{ [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] }

Let *X* be an algebraic curve ^{ [8] } given by an affine equation 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 .

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 (since *K* 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 *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 (whose normalization is a line with coordinate *t* and map ) 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 and 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 (the *x* coordinate ramifies at this point, so it is not a power series).

The curve (whose normalization is again a line with coordinate *t* and map ), on the other hand, has a single branch at the cusp point (0,0), 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 *n*-th root of *x*, they converge for small enough , 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*).

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 *A* for any given bound *A*. 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,^{ [12] }^{ [13] } 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.^{ [14] }

- ↑ Newton (1960)
- 1 2 Puiseux (1850, 1851)
- ↑ Newton (1736)
- 1 2 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)

An **algebraic number** is any complex number that is a root of a non-zero polynomial in one variable with rational coefficients.

In mathematics, the **binomial coefficients** are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers *n* ≥ *k* ≥ 0 and is written It is the coefficient of the *x*^{k} term in the polynomial expansion of the binomial power (1 + *x*)^{n}, and is given by the formula

In elementary algebra, the **binomial theorem** describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (*x* + *y*)^{n} into a sum involving terms of the form *ax*^{b}*y*^{c}, where the exponents b and c are nonnegative integers with *b* + *c* = *n*, and the coefficient a of each term is a specific positive integer depending on n and b. For example,

In mathematics, a **polynomial** is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate *x* is *x*^{2} − 4*x* + 7. An example in three variables is *x*^{3} + 2*xyz*^{2} − *yz* + 1.

The **fundamental theorem of algebra** states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.

In mathematics, a **formal power series** is a generalization of a polynomial, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the powers of the variable are used only as position-holders for the coefficients, so that the coefficient of is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved. More generally, formal power series can include series with any finite number of variables, and with coefficients in an arbitrary ring.

In mathematics, **factorization** or **factoring** consists of writing a number or another mathematical object as a product of several *factors*, usually smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, and (*x* – 2)(*x* + 2) is a factorization of the polynomial *x*^{2} – 4.

In algebra, the **partial fraction decomposition** or **partial fraction expansion** of a rational fraction is an operation that consists of expressing the fraction as a sum of a polynomial and one or several fractions with a simpler denominator.

In mathematics, an **affine algebraic plane curve** is the zero set of a polynomial in two variables. A **projective algebraic plane curve** is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation *h*(*x*, *y*, *t*) = 0 can be restricted to the affine algebraic plane curve of equation *h*(*x*, *y*, 1) = 0. These two operations are each inverse to the other; therefore, the phrase **algebraic plane curve** is often used without specifying explicitly whether it is the affine or the projective case that is considered.

In algebra, a **valuation** is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a **valued field**.

In transcendental number theory, the **Lindemann–Weierstrass theorem** is a result that is very useful in establishing the transcendence of numbers. It states the following.

In mathematics, a **monomial** is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:

- A monomial, also called
**power product**, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, is a monomial. The constant 1 is a monomial, being equal to the empty product and to x^{0}for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power*x*^{n}of x, with n a positive integer. If several variables are considered, say, then each can be given an exponent, so that any monomial is of the form with non-negative integers. - A monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A monomial in the first sense is a special case of a monomial in the second sense, where the coefficient is 1. For example, in this interpretation and are monomials.

In mathematics, especially in the field of algebra, a **polynomial ring** or **polynomial algebra** is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In mathematics, the **binomial series** is the Taylor series for the function given by where is an arbitrary complex number and |*x*| < 1. Explicitly,

In mathematics, the **Newton polygon** is a tool for understanding the behaviour of polynomials over local fields.

In mathematics, **tropical geometry** is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition:

In mathematics, an **algebraic function** is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:

In mathematics, **Hahn series** are a type of formal infinite series. They are a generalization of Puiseux series and were first introduced by Hans Hahn in 1907. They allow for arbitrary exponents of the indeterminate so long as the set supporting them forms a well-ordered subset of the value group. Hahn series were first introduced, as groups, in the course of the proof of the Hahn embedding theorem and then studied by him as fields in his approach to Hilbert's seventeenth problem.

In mathematics, the **Levi-Civita field**, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Each member can be constructed as a formal series of the form

In mathematics, an **algebraic number field** is an extension field of the field of rational numbers such that the field extension has finite degree . Thus is a field that contains and has finite dimension when considered as a vector space over .

- 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.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.