nLab
Puiseux series

Definition

Let $K$ be a field. A Puiseux series with coefficients in $K$ is a formal Laurent series

$$f(t)=\sum _{k\ge m}{a}_k{t}^{k/r}(1)$$f(t) = \sum_{k \geq m} a_k t^{k/r} \qquad (1)

where $r$ is a positive integer, $m$ is an integer, and each $a_k$ belongs to $K$. Somewhat more abstractly but also more meaningfully, if ${\mathbb{N}}_{\cdot }$ is the poset of positive integers ordered by divisibility (so that $1$ is the least element), then the field of Puiseux series is the filtered colimit of the diagram of fields

$$F:{\mathbb{N}}_{\cdot }\to \mathrm{Field}$$F: \mathbb{N}_{\cdot} \to Field

where each $F(n)$ is the field of Laurent series $K((t))$, but where $F(m)\to F(n)$ (in case $m\mid n$) is the field homomorphism taking $f(t)$ to $f(t^{n/m})$.

The field of Puiseux series carries a valuation whose values are in the ordered group of rationals $(\mathbb{Q},+)$: for $f(t)$ as in (1), $v(f)$ is the least exponent $k/r$ for which $a_k\ne 0$.

Puiseux-Newton expansions

Puiseux series were in essence considered by Isaac Newton, who developed a method of expanding algebraic functions as Puiseux series, based on an analogue of Newton's method of approximating roots. Here is a sample theorem:

Theorem

If $K$ is algebraically closed and has characteristic 0, then the field of Puiseux series over $K$ is the algebraic closure of the field of Laurent series over $K$.

Proof (sketch)

It is enough to show that every degree $n$ extension $E$ of the field of Laurent series $K((x))$ is of the form $K((x^{1/n}))$. For this, it suffices that the integral closure $B$ of $K[[x]]$ in $E$ be of the form $K[[x^{1/n}]]$.

Generally speaking, let $A$ be a complete DVR (discrete valuation ring) with maximal ideal ${𝔪}_A$ and residue class field $k_A$, and let $F$ be its field of fractions. Let $E$ be a degree $n$ extension of $F$, and let $B$ be the integral closure of $A$ in $E$. Then $B$ is also a complete DVR. We may write the ideal ${𝔪}_{A}B$ of $B$ as ${𝔪}_B^e$ where $e$ is the ramification index, and we have

$$n={\mathrm{deg}}_{F}E={\mathrm{rank}}_{A}B={\mathrm{rank}}_{A/{𝔪}_A}B/{𝔪}_{A}B={\dim }_{k_A}B/{𝔪}_B^e=e\cdot {\dim }_{k_A}{k}_B$$n = deg_F E = rank_A B = rank_{A/\mathfrak{m}_A} B/\mathfrak{m}_A B = dim_{k_A} B/\mathfrak{m}_B^e = e \cdot dim_{k_A} k_B

where the last equation holds because ${𝔪}_B^i/{𝔪}_B^{i+1}\cong k_B$ as $k_B$-modules and therefore also as $k_A$-modules. In the case $A=K[[x]]$ where $k_A=K$, we have that ${\dim }_{k_A}{k}_B=1$ since $K$ is algebraically closed, and therefore $e=n$. In other words, $(x)B={𝔪}_B^n$, so we can write $x=u{\pi }^n$ where $\pi$ generates the maximal ideal of $B$ and $u$ is a unit of $B$.

The residue class $\overline{u}\in K$ has an $n^{\mathrm{th}}$ root (again by algebraic closure); in fact a simple root since $\mathrm{char}(K)=0$. By Hensel's lemma?, this lifts to an $n^{\mathrm{th}}$ root of $u$ in $B$. The element $u^{1/n}\pi$ is thus an $n^{\mathrm{th}}$ root of $x$, and is a generator of the maximal ideal of $B$. Writing this element as $y=x^{1/n}$, the ring $K[[y]]=A[x^{1/n}]\hookrightarrow B$ is an $A$-submodule of full rank $n$ and integrally closed (being abstractly isomorphic to $A$, which is integrally closed because it’s a principal ideal domain and therefore a unique factorization domain), so that $K[[y]]=B$, as was to be shown.

Example

(Intend to solve for $y$ in $y^3-xy+1=0$ as a Puiseux series in $x$.)

References

• wikipedia Puiseux series

The sketched proof of theorem 1 was extracted from notes on a seminar by Boyarchenko on local class field theory:

• Mitya Boyarchenko, Kottwitz Seminar Lectures, (U. Michigan, Winter 2011). (pdf)

and the reader may refer to the classic text by Serre for a fuller treatment:

• Jean-Pierre Serre, Local Fields (trans. Marvin J. Greenberg), Graduate Texts in Math. 67, Springer 1980.

For a noncommutative generalization see

• D. Grigoriev, Analogue of Newton-Puiseux series for non-holonomic D-modules and factoring, Moscow Math. J. 9, 2009, 775–800, pdf; MPIM2008-6, pdf

Other references

• Jan Kiwi, Puiseux series dynamics of Quadratic rational maps, arxiv/1106.0059

(Proof or suitable reference to be inserted later.)