Puiseux series
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Puiseux series are a generalization of
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
that allow for negative and fractional exponents of the indeterminate. For example, the series : \begin x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\ &=x^+ 2x^ + x^ + 2x^ + x^ + x^ + \cdots \end is a Puiseux series in the indeterminate . Puiseux series were first introduced by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
in 1676 and rediscovered by Victor Puiseux in 1850.Puiseux (1850, 1851) The definition of a Puiseux series includes that the denominators of the exponents must be bounded. So, by reducing exponents to a common denominator , a Puiseux series becomes a
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
in a th root of the indeterminate. For example, the example above is a Laurent series in x^. Because a complex number has th roots, a convergent Puiseux series typically defines functions in a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of . Puiseux's theorem, sometimes also called the Newton–Puiseux theorem, asserts that, given a
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
P(x,y)=0 with complex coefficients, its solutions in , viewed as functions of , may be expanded as Puiseux series in that are convergent in some
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
of . In other words, every branch of an
algebraic curve 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 ...
may be locally described by a Puiseux series in (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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
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 In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of the
ring of formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
.


Definition

If is a field (such as the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s), a ''Puiseux series'' with coefficients in is an expression of the form :f = \sum_^ c_k T^ where n is a positive integer and k_0 is an integer. In other words, Puiseux series differ from
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
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^ + 2T^ + T^ + \cdots) + (T^ - T^ + 2 + \cdots) = T^ + T^ + T^ + 2 + \cdots and : (T^ + 2T^ + T^ + \cdots) \cdot (T^ - T^ + 2 + \cdots) = T^ + 2T^ - T^ + T^ + 4T^ + \cdots. 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 T^. The Puiseux series with coefficients in form a field, which is the union :\bigcup_ K(\!(T^)\!) of fields of
formal Laurent series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
in T^ (considered as an indeterminate). This yields an alternative definition of the field of Puiseux series in terms of a
direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
. For every positive integer , let T_n be an indeterminate (meant to represent T^), and K(\!(T_n)\!) be the field of formal Laurent series in T_n. If divides , the mapping T_m \mapsto (T_n)^ induces a
field homomorphism Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) Definition of a field A field is a commutative rin ...
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 Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
an isomorphism).


Valuation

A nonzero Puiseux series f can be uniquely written as :f = \sum_^ c_k T^ with c_\neq 0. The ''valuation'' :v(f) = \frac n of f is the smallest exponent for the natural order of the rational numbers, and the corresponding coefficient c_ is called the ''initial coefficient'' or ''valuation coefficient'' of f. The valuation of the zero series is +\infty. The function is a valuation and makes the Puiseux series a
valued field Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...
, with the additive group \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
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
. The notation :f = \sum_^ c_k T^ expresses that a Puiseux is the limit of its partial sums. However, the field of Puiseux series is not
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
; see below .


Convergent Puiseux series

Puiseux series provided by Newton–Puiseux theorem are convergent in the sense that there is a neighborhood of zero in which they are convergent (0 excluded if the valuation is negative). More precisely, let :f = \sum_^ c_k T^ be a Puiseux series with
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
coefficients. There is a real number , called the
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
such that the series converges if is substituted for a nonzero complex number of absolute value less than , and 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 th roots, some care must be taken for the substitution: a specific th root of , say , must be chosen. Then the substitution consists of replacing T^ by x^k for every . The existence of the radius of convergence results from the similar existence for a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
, applied to T^f, considered as a power series in T^. 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 mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
in some neighborhood of zero (zero itself possibly excluded).


Valuation and order on coefficients

If the base field K is ordered, then the field of Puiseux series over K is also naturally (“
lexicographically In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...
”) 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 \omega is infinitely large (in other words, the value group of \hat w is \Q \times \Gamma ordered lexicographically, where \Gamma 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

As early as 1671,
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
implicitly used Puiseux series and proved the following theorem for approximating with series the
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
of
algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s whose coefficients are functions that are themselves approximated with series or
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s. 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 introduced the concept of (non-truncated) Puiseux series and proved the theorem that is now known as ''Puiseux's theorem'' or ''Newton–Puiseux theorem''.cf. Kedlaya (2001), introduction The theorem asserts that, given an algebraic equation whose coefficients are polynomials or, more generally, Puiseux series over a field of
characteristic zero In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
, 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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, 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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
''.


Newton polygon

Let :P(y)=\sum_ a_i(x) y^i be a polynomial whose nonzero coefficients a_i(x) are polynomials, power series, or even Puiseux series in . In this section, the valuation v(a_i) of a_i is the lowest exponent of in a_i. (Most of what follows applies more generally to coefficients in any valued ring.) For computing the Puiseux series that are
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
of (that is solutions of the
functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
P(y)=0), the first thing to do is to compute the valuation of the roots. This is the role of the Newton polygon. Let consider, in a
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, the points of coordinates (i, v(a_i)). The ''Newton polygon'' of is the lower
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of these points. That is, the edges of the Newton polygon are the
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
s joigning two of these points, such that all these points are not below the line supporting the segment (below is, as usually, relative to the value of the second coordinate). Given a Puiseux series y_0 of valuation v_0, the valuation of P(y_0) is at least the minimum of the numbers i v_0 + v(a_i), and is equal to this minimum if this minimum is reached for only one . So, for y_0 being a root of , the minimum must be reached at least twice. That is, there must be two values i_1 and i_2 of such that i_1 v_0 + v(a_) = i_2 v_0 + v(a_), and i v_0 + v(a_) \ge i_1 v_0 + v(a_) for every . That is, (i_1, v(a_)) and (i_2, v(a_)) must belong to an edge of the Newton polygon, and v_0=-\frac must be the opposite of the slope of this edge. This is a rational number as soon as all valuations v(a_i) are rational numbers, and this is the reason for introducing rational exponents in Puiseux series. In summary, ''the valuation of a root of'' ''must be the opposite of a slope of an edge of the Newton polynomial.'' The initial coefficient of a Puiseux series solution of P(y)=0 can easily be deduced. Let c_i be the initial coefficient of a_i(x), that is, the coefficient of x^ in a_i(x). Let -v_0 be a slope of the Newton polygon, and \gamma x_0^ be the initial term of a corresponding Puiseux series solution of P(y)=0. If no cancellation would occur, then the initial coefficient of P(y) would be \sum_c_i \gamma^i, where is the set of the indices such that (i, v(a_i)) belongs to the edge of slope v_0 of the Newton polygon. So, for having a root, the initial coefficient \gamma must be a nonzero root of the polynomial \chi(x)=\sum_c_i x^i (this notation will be used in the next section). In summary, the Newton polynomial allows an easy computation of all possible initial terms of Puiseux series that are solutions of P(y)=0. The proof of Newton–Puiseux theorem will consist of starting from these initial terms for computing recursively the next terms of the Puiseux series solutions.


Constructive proof

Let suppose that the first term \gamma x^ of a Puiseux series solution of P(y)=0 has been be computed by the method of the preceding section. It remains to compute z=y-\gamma x^. For this, we set y_0=\gamma x^, and write the
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
of at z=y-y_0: :Q(z)=P(y_0+z)=P(y_0)+zP'(y_0)+\cdots + z^j\frac +\cdots This is a polynomial in whose coefficients are Puiseux series in . One may apply to it the method of the Newton polygon, and iterate for getting the terms of the Puiseux series, one after the other. But some care is required for insuring that v(z)>v_0, and showing that one get a Puiseux series, that is, that the denominators of the exponents of remain bounded. The derivation with respect to does not change the valuation in of the coefficients; that is, :v\left(P^(y_0)z^j\right)\ge \min_i (v(a_i) +v_0)+j(v(z)-v_0), and the equality occurs if and only if \chi^(\gamma)\neq 0, where \chi(x) is the polynomial of the preceding section. If is the multiplicity of \gamma as a root of \chi, it results that the inequality is an equality for j=m. The terms such that j>m can be forgotten as far as one is concerned by valuations, as v(z)>v_0 and j>m imply :v\left(P^(y_0)z^j\right) \ge \min_i (v(a_i) +iv_0)+j(v(z)-v_0) > v\left(P^(y_0)z^m\right). This means that, for iterating the method of Newton polygon, one can and one must consider only the part of the Newton polygon whose first coordinates belongs to the interval
, m The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
Two cases have to be considered separately and will be the subject of next subsections, the so-called ''ramified case'', where , and the ''regular case'' where .


Regular case


Ramified case

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 P(y) in . Without loss of generality, one can suppose that P(0)\neq 0, that is, a_0\neq 0. Indeed, each factor of P(y) provides a solution that is the zero Puiseux series, and such factors can be factored out. As the charactistic is supposed to be zero, one can also suppose that P(y) is a square-free polynomial, that is that the solutions of P(y)=0 are all different. Indeed, the square-free factorization uses only the operations of the field of coefficients for factoring P(y) into square-free factors than can be solved separately. (The hypothesis of characteristic zero is needed, since, in characteristic , the square-free decomposition can provide irreducible factors, such as y^p-x, 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 P(0)\neq 0, the length of the Newton polygon of is its degree in , 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 \chi(x). 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 \chi(x) that is square free, and the computation can continue as in the regular case for each root of \chi(x). 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.


Failure in positive characteristic

The Newton–Puiseux theorem is not valid over fields of positive characteristic. For example, the equation X^2 - X = T^ has solutions :X = T^ + \frac + \fracT^ - \fracT^ + \cdots and :X = -T^ + \frac - \fracT^ + \fracT^ + \cdots (one readily checks on the first few terms that the sum and product of these two series are 1 and -T^ 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^ shows this: reasoning with valuations shows that ''X'' should have valuation -\frac, and if we rewrite it as X = T^ + X_1 then :X^p = T^ + ^p,\text^p - X_1 = T^ and one shows similarly that X_1 should have valuation -\frac, and proceeding in that way one obtains the series :T^ + T^ + T^ + \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)\!). Similarly to the case of algebraic closure, there is an analogous theorem for
real closure In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. D ...
: 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 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 ...
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 = c t^k + \cdots: then there is a unique Puiseux series of the form f = c T^ + \cdots (a power series in T^), such that y(t)=f(x(t)) (the latter expression is meaningful since x(t)^ = 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. 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 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 + \fracx^2 - \fracx^3 + \cdots and y = - x - \fracx^2 + \fracx^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)^ + (x+1)^ (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^.


Analytic convergence

When K=\Complex 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


Levi-Civita field

The field of Puiseux series is not
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
as a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
. Its completion, called the
Levi-Civita field 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 a can be constructed as a formal series of the form ...
, 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 +\infty. 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_^ T^ is not a Puiseux series, but it is the limit of a
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
of Puiseux series; in particular, it is the limit of \sum_^ T^ 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, hence the opportunity of completing it even more.


Hahn series

Hahn series In mathematics, Hahn series (sometimes also known as Hahn–Mal'cev–Neumann series) are a type of formal infinite series. They are a generalization of Puiseux series (themselves a generalization of formal power series) and were first introduced ...
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 Hilbert's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It concerns the expression of positive definite rational functions as sums of quotients of squares. The original q ...
. 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 \Q or \R). These were later further generalized by
Anatoly Maltsev Anatoly Ivanovich Maltsev (also: Malcev, Mal'cev; Russian: Анато́лий Ива́нович Ма́льцев; 27 November N.S./14 November O.S. 1909, Moscow Governorate – 7 June 1967, Novosibirsk) was born in Misheronsky, near Moscow, an ...
and
Bernhard Neumann Bernhard Hermann Neumann (15 October 1909 – 21 October 2002) was a German-born British-Australian mathematician, who was a leader in the study of group theory. Early life and education After gaining a D.Phil. from Friedrich-Wilhelms Universit ...
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.Kedlaya (2001)


Notes


See also

*
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
*
Madhava series In mathematics, a Madhava series or Leibniz series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by an Indian Mathematician and Astronomer Madhava of Sangamagrama (c.&nbs ...
* Newton's divided difference interpolation *
Padé approximant In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is ap ...


References

* * * * * * (Translated from Latin) * * * * *


External links

*
Puiseux series at MathWorld




{{series (mathematics) Commutative algebra Algebraic curves Mathematical series