In mathematics, Puiseux series are a generalization of power series, first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850, that allow for negative and fractional exponents of the indeterminate T. A Puiseux series in the indeterminate T is a Laurent series in T1/n, where n is a positive integer. A Puiseux series may be written as:
Contents
- Field of Puiseux series
- Valuation and order
- Algebraic closedness of Puiseux series
- Algebraic curves
- Analytic convergence
- Generalization
- References
where
Puiseux's theorem, sometimes also called Newton–Puiseux theorem, asserts that, given a polynomial equation
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 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.
Field of Puiseux series
If K is a field then we can define the field of Puiseux series with coefficients in K (or over K) informally as the set of formal expressions of the form
where
In other words, the field of Puiseux series with coefficients in K is the union of the fields
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
Valuation and order
The Puiseux series over a field K form a valued field with value group
as above is defined to be the smallest rational
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
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
Algebraic closedness of Puiseux series
One essential property of Puiseux series is expressed by the following theorem, attributed to Puiseux (for
Theorem: if K is an algebraically closed field of characteristic zero, then the field of Puiseux series over K is the algebraic closure of the field of formal Laurent series over K.
Very roughly, the proof proceeds essentially by inspecting the Newton polygon of the equation and extracting the coefficients one by one using a valuative form of Newton's method. Provided algebraic equations can be solved algorithmically in the base field K, then the coefficients of the Puiseux series solutions can be computed to any given order.
For example, the equation
and
(one readily checks on the first few terms that the sum and product of these two series are 1 and
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
and one shows similarly that
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
Similarly to the case of algebraic closure, there is an analogous theorem for real closure: if K is a real closed field, then the field of Puiseux series over K is the real closure of the field of formal Laurent series over K. (This implies the former theorem since any algebraically closed field of characteristic zero is the unique quadratic extension of some real-closed field.)
There is also an analogous result for p-adic closure: if K is a p-adically closed field with respect to a valuation w, then the field of Puiseux series over K is also p-adically closed.
Algebraic curves
Let X be an algebraic curve given by an affine equation
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
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
The curve
Analytic convergence
When
Generalization
The field of Puiseux series is not complete, its completion can be described as follows: it is the field of formal expressions of the form
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), where instead of requiring the exponents to have bounded denominator they are required to form a well-ordered subset of the value group (usually