Puiseux series

The Puiseux series over a field K form a valued field with value group ${\displaystyle \mathbb {Q} }$ (the rationals): the valuation ${\displaystyle v(f)}$ of a series

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

as above is defined to be the smallest rational ${\displaystyle k/n}$ such that the coefficient ${\displaystyle c_{k}}$ of the term with exponent ${\displaystyle k/n}$ is non-zero (with the usual convention that the valuation of 0 is +∞). The coefficient ${\displaystyle c_{k}}$ 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 ${\displaystyle \exp(-v(f))}$. This justifies a posteriori the notation

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

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

One essential property of Puiseux series is expressed by the following theorem, attributed to Puiseux[2] (for ${\displaystyle K=\mathbb {C} }$) 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 ${\displaystyle X^{2}-X=T^{-1}}$ has solutions

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

and

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

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

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

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

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

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 ${\displaystyle K(\!(T)\!)}$.[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]

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

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 ${\displaystyle x=t^{n}+\cdots }$ (since K is algebraically closed, we can assume the valuation coefficient to be 1) and ${\displaystyle y=ct^{k}+\cdots }$: then there is a unique Puiseux series of the form ${\displaystyle f=cT^{k/n}+\cdots }$ (a power series in ${\displaystyle T^{1/n}}$), such that ${\displaystyle y(t)=f(x(t))}$ (the latter expression is meaningful since ${\displaystyle x(t)^{1/n}=t+\cdots }$ 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 ${\displaystyle y^{2}=x^{3}+x^{2}}$ (whose normalization is a line with coordinate t and map ${\displaystyle t\mapsto (t^{2}-1,t^{3}-t)}$) 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 ${\displaystyle y=x+{\frac {1}{2}}x^{2}-{\frac {1}{8}}x^{3}+\cdots }$ and ${\displaystyle y=-x-{\frac {1}{2}}x^{2}+{\frac {1}{8}}x^{3}+\cdots }$ 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 ${\displaystyle y=-(x+1)^{1/2}+(x+1)^{3/2}}$ (the x coordinate ramifies at this point, so it is not a power series).

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

When ${\displaystyle K=\mathbb {C} }$ 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 ${\displaystyle |x|}$, 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).

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 ${\displaystyle f=\sum _{e}c_{e}T^{e},}$ where the support of the coefficients (that is, the set of e such that ${\displaystyle c_{e}\neq 0}$) 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, ${\displaystyle \sum _{k=1}^{+\infty }T^{k+{\frac {1}{k}}}}$ is not a Puiseux series, but it is the limit of a Cauchy sequence of Puiseux series; in particular, it is the limit of ${\displaystyle \sum _{k=1}^{N}T^{k+{\frac {1}{k}}}}$ as ${\displaystyle N\to +\infty }$. 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 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 ${\displaystyle \mathbb {Q} }$ or ${\displaystyle \mathbb {R} }$). 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]