Hahn series


In mathematics, Hahn series are a type of formal infinite series. They are a generalization of Puiseux series and were first introduced by Hans Hahn in 1907. They allow for arbitrary exponents of the indeterminate so long as the set supporting them forms a well-ordered subset of the value group. Hahn series were first introduced, as groups, in the course of the proof of the Hahn embedding theorem and then studied by him in relation to Hilbert's second problem.

Formulation

The field of Hahn series over a field and with value group is the set of formal expressions of the form
with such that the support of f is well-ordered. The sum and product of
are given by
and
.
For example, is a Hahn series because the set of rationals
is well-ordered; it is not a Puiseux series because the denominators in the exponents are unbounded.

Properties

Properties of the valued field

The valuation of a non-zero Hahn series
is defined as the smallest such that : this makes into a spherically complete valued field with value group and residue field . In fact, if has characteristic zero, then is up to isomorphism the only spherically complete valued field with residue field and value group.
The valuation defines a topology on. If, then corresponds to an ultrametric absolute value, with respect to which is a complete metric space. However, unlike in the case of formal Laurent series or Puiseux series, the formal sums used in defining the elements of the field do not converge: in the case of for example, the absolute values of the terms tend to 1, so the series is not convergent.

Algebraic properties

If is algebraically closed and is divisible, then is algebraically closed. Thus, the algebraic closure of is contained in, where is the algebraic closure of : in fact, it is possible to give a somewhat analogous description of the algebraic closure of in positive characteristic as a subset of.
If is an ordered field then is totally ordered by making the indeterminate infinitesimal or, equivalently, by using the lexicographic order on the coefficients of the series. If is real-closed and is divisible then is itself real-closed. This fact can be used to analyse the field of surreal numbers.
If κ is an infinite regular cardinal, one can consider the subset of consisting of series whose support set has cardinality less than κ: it turns out that this is also a field, with much the same algebraic closedness properties as the full : e.g., it is algebraically closed or real closed when is so and is divisible.

Summable families

Summable families

One can define a notion of summable families in. If is a set and is a family of Hahn series, then we say that is summable if the set is well-ordered, and each set for is finite.
We may then define the sum as the Hahn series
If are summable, then so are the families, and we have
and
This notion of summable family does not correspond to the notion of convergence in the valuation topology on. For instance, in, the family is summable but the sequence does not converge.

Evaluating analytic functions

Let and let denote the ring of real-valued functions which are analytic on a neighborhood of.
If contains, then we can evaluate every element of at every element of of the form, where the valuation of is strictly positive.
Indeed, the family is always summable, so we can define. This defines a ring homomorphism.

Hahn–Witt series

The construction of Hahn series can be combined with Witt vectors to form twisted Hahn series or Hahn–Witt series: for example, over a finite field K of characteristic p, the field of Hahn–Witt series with value group Γ would be the set of formal sums where now are Teichmüller representatives which are multiplied and added in the same way as in the case of ordinary Witt vectors. When Γ is the group of rationals or reals and K is the algebraic closure of the finite field with p elements, this construction gives a metrically complete algebraically closed field containing the p-adics, hence a more or less explicit description of the field or its spherical completion.

Examples

  • The field of formal Laurent series over can be described as.
  • The field of surreal numbers can be regarded as a field of Hahn series with real coefficients and value group the surreal numbers themselves.
  • The Levi-Civita field can be regarded as a subfield of, with the additional imposition that the coefficients be a left-finite set: the set of coefficients less than a given coefficient is finite.
  • The field of transseries is a directed union of Hahn fields. The construction of resembles ,.