Hahn series

The valuation ${\displaystyle v(f)}$ of a non zero Hahn series

${\displaystyle f=\sum _{e\in \Gamma }c_{e}T^{e}}$

is defined as the smallest ${\displaystyle e\in \Gamma }$ such that ${\displaystyle c_{e}\neq 0}$ (in other words, the smallest element of the support of ${\displaystyle f}$): this makes ${\displaystyle K[[T^{\Gamma }]]}$ into a spherically complete valued field with value group ${\displaystyle \Gamma }$ and residue field ${\displaystyle K}$ (justifying a posteriori the terminology). In fact, if ${\displaystyle K}$ has characteristic zero, then ${\displaystyle (K[[T^{\Gamma }]],v)}$ is up to (non unique) isomorphism the only spherically complete valued field with residue field ${\displaystyle K}$ and value group ${\displaystyle \Gamma }$.[2] The valuation ${\displaystyle v}$ defines a topology on ${\displaystyle K\left[\left[T^{\Gamma }\right]\right]}$. If ${\displaystyle \Gamma \subseteq \mathbb {R} }$, then v corresponds to an ultrametric absolute value ${\displaystyle |f|=\exp(-v(f))}$, with respect to which ${\displaystyle K\left[\left[T^{\Gamma }\right]\right]}$ 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 ${\displaystyle T^{-1/p}+T^{-1/p^{2}}+T^{-1/p^{3}}+\cdots }$ for example, the absolute values of the terms tend to 1 (because their valuations tend to 0), so the series is not convergent (such series are sometimes known as "pseudo-convergent"[3]).

If K is algebraically closed (but not necessarily of characteristic zero) and Γ is divisible, then ${\displaystyle K\left[\left[T^{\Gamma }\right]\right]}$ is algebraically closed.[4] Thus, the algebraic closure of ${\displaystyle K((T))}$ is contained in ${\displaystyle {\overline {K}}[[T^{\mathbb {Q} }]]}$, where ${\displaystyle {\overline {K}}}$ is the algebraic closure of ${\displaystyle K}$ (when K is of characteristic zero, it is exactly the field of Puiseux series): in fact, it is possible to give a somewhat analogous description of the algebraic closure of ${\displaystyle K((T))}$ in positive characteristic as a subset of ${\displaystyle K\left[\left[T^{\Gamma }\right]\right]}$.[5]

If K is an ordered field then ${\displaystyle K\left[\left[T^{\Gamma }\right]\right]}$ is totally ordered by making the indeterminate T infinitesimal (greater than 0 but less than any positive element of K) or, equivalently, by using the lexicographic order on the coefficients of the series. If K is real-closed and Γ is divisible then ${\displaystyle K\left[\left[T^{\Gamma }\right]\right]}$ is itself real closed.[6] This fact can be used to analyse (or even construct) the field of surreal numbers (which is isomorphic, as an ordered field, to the field of Hahn series with real coefficients and value group the surreal numbers themselves[7]).

If κ is an infinite regular cardinal, one can consider the subset of ${\displaystyle K\left[\left[T^{\Gamma }\right]\right]}$ consisting of series whose support set ${\displaystyle \{e\in \Gamma :c_{e}\neq 0\}}$ has cardinality (strictly) less than κ: it turns out that this is also a field, with much the same algebraic closedness properties as the full ${\displaystyle K\left[\left[T^{\Gamma }\right]\right]}$: e.g., it is algebraically closed or real closed when K is so and Γ is divisible.[8]

One can define a notion of summable families in ${\displaystyle K\left[\left[T^{\Gamma }\right]\right]}$. If ${\displaystyle I}$ is a set and ${\displaystyle (f_{i})_{i\in I}}$ is a family of Hahn series ${\displaystyle f_{i}\in K\left[\left[T^{\Gamma }\right]\right]}$, then we say that ${\displaystyle (f_{i})_{i\in I}}$ is summable if the set ${\displaystyle \bigcup \limits _{i\in I}\operatorname {supp} f_{i}\subset \Gamma }$ is well-ordered, and each set ${\displaystyle \{i\in I\ |\ e\in \operatorname {supp} f_{i}\}}$ for ${\displaystyle e\in \Gamma }$ is finite.

We may then define the sum ${\displaystyle \sum \limits _{i\in I}f_{i}}$ as the Hahn series

${\displaystyle \sum \limits _{i\in I}f_{i}:=\sum \limits _{e\in \Gamma }(\sum \limits _{i\in I}f_{i}(e))T^{e}}$

.

If ${\displaystyle (f_{i})_{i\in I},(g_{i})_{i\in I}}$ are summable, then so are the families ${\displaystyle (f_{i}+g_{i})_{i\in I},(f_{i}g_{j})_{(i,j)\in I\times I}}$, and we have[9]

${\displaystyle \sum \limits _{i\in I}f_{i}+g_{i}=\sum \limits _{i\in I}f_{i}+\sum \limits _{i\in I}g_{i}}$

and

${\displaystyle \sum \limits _{(i,j)\in I\times I}f_{i}g_{j}=(\sum \limits _{i\in I}f_{i})(\sum \limits _{i\in I}g_{i})}$

.

This notion of summable family does not correspond to the notion of convergence in the valuation topology on ${\displaystyle K\left[\left[T^{\Gamma }\right]\right]}$. For instance, in ${\displaystyle \mathbb {Q} \left[\left[T^{\mathbb {Q} }\right]\right]}$, the family ${\displaystyle (T^{\frac {n}{n+1}}+T^{n+1})_{n\in \mathbb {N} }}$ is summable but the sequence ${\displaystyle (\sum \limits _{k\leq n}T^{\frac {k}{k+1}}+T^{k+1})_{n\in \mathbb {N} }}$ does not converge.

Let ${\displaystyle a\in \mathbb {R} }$< and let ${\displaystyle {\mathcal {A}}_{a}}$ denote the ring of real-valued functions which are analytic on a neighborhood of ${\displaystyle a}$.

If ${\displaystyle K}$ contains ${\displaystyle \mathbb {R} }$, then we can evaluate every element ${\displaystyle f}$ of ${\displaystyle {\mathcal {A}}_{a}}$ at every element of ${\displaystyle K\left[\left[T^{\Gamma }\right]\right]}$ of the form ${\displaystyle a+\varepsilon }$, where the valuation of ${\displaystyle \varepsilon }$ is strictly positive. Indeed, the family ${\displaystyle ({\frac {f^{(n)}(a)}{n!}}\varepsilon ^{n})_{n\in \mathbb {N} }}$ is always summable,[10] so we can define ${\displaystyle f(a+\varepsilon ):=\sum \limits _{n\in \mathbb {N} }{\frac {f^{(n)}(a)}{n!}}\varepsilon ^{n}}$. This defines a ring morphism ${\displaystyle {\mathcal {A}}_{a}\longrightarrow K\left[\left[T^{\Gamma }\right]\right]}$.