Puiseux series



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

f(t)= kma kt k/r(1)f(t) = \sum_{k \geq m} a_k t^{k/r} \qquad (1)

where rr is a positive integer, mm is an integer, and each a ka_k belongs to KK. Somewhat more abstractly but also more meaningfully, if \mathbb{N}_{\cdot} is the poset of positive integers ordered by divisibility (so that 11 is the least element), then the field of Puiseux series is the filtered colimit of the diagram of fields

F: FieldF: \mathbb{N}_{\cdot} \to Field

where each F(n)F(n) is the field of Laurent series K((t))K((t)), but where F(m)F(n)F(m) \to F(n) (in case mnm|n) is the field homomorphism taking f(t)f(t) to f(t n/m)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)f(t) as in (1), v(f)v(f) is the least exponent k/rk/r for which a k0a_k \neq 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:


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

Proof (sketch)

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

Generally speaking, let AA be a complete DVR (discrete valuation ring) with maximal ideal 𝔪 A\mathfrak{m}_A and residue class field k Ak_A, and let FF be its field of fractions. Let EE be a degree nn extension of FF, and let BB be the integral closure of AA in EE. Then BB is also a complete DVR. We may write the ideal 𝔪 AB\mathfrak{m}_A B of BB as 𝔪 B e\mathfrak{m}_B^e where ee is the ramification index, and we have

n=deg FE=rank AB=rank A/𝔪 AB/𝔪 AB=dim k AB/𝔪 B e=edim k Ak Bn = 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+1k B\mathfrak{m}_B^i/\mathfrak{m}_B^{i+1} \cong k_B as k Bk_B-modules and therefore also as k Ak_A-modules. In the case A=K[[x]]A = K[ [x] ] where k A=Kk_A = K, we have that dim k Ak B=1dim_{k_A} k_B = 1 since KK is algebraically closed, and therefore e=ne = n. In other words, (x)B=𝔪 B n(x)B = \mathfrak{m}_B^n, so we can write x=uπ nx = u \pi^n where π\pi generates the maximal ideal of BB and uu is a unit of BB.

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


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


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
  • Luis Felipe Tabera, On real tropical bases and real tropical discriminants, arxiv/1311.2211

We explore the concept of real tropical basis of an ideal in the field of real Puiseux series. We show explicit tropical bases of zero-dimensional real radical ideals, linear ideals and hypersurfaces coming from combinatorial patchworking. But we also show that there exist real radical ideals that do not admit a tropical basis. As an application, we show how to compute the set of singular points of a real tropical hypersurface. i.e. we compute the real tropical discriminant.

Revised on November 12, 2013 04:00:46 by Zoran Škoda (