nLab Hahn series

Idea

A Hahn series is a generalization of a formal power series which allows non-integer exponents and transfinite sums, as long as the exponents are well-ordered.

Definition

Let $G$ be a linearly ordered abelian group, and $k$ a field.

Definition

The ring of Hahn series with value group? $G$, denoted $k[t^G]$ (or sometimes $k[[t^G]]$ or $k((t^G))$), is the ring of functions $f\colon G \to k$ such that $\{x \in G : f(x) \neq 0\}$ is well-ordered as as a subset of $G$ (or, sometimes, as a subset of the opposite poset $G^{op}$). Addition is defined pointwise, and multiplication is defined by the convolution product:

$$(f \cdot g)(x) = \sum_{y+z = x \in G} f(y)g(z)$$

Notationally, we may write a Hahn series $f\colon G \to k$ as $\sum_{x\in G} f(x) t^x$.

Properties

Theorem

The ring $k[t^G]$ is a field. If $k$ is algebraically closed, then $k[t^G]$ is algebraically closed provided that $G$ is divisible.

As a corollary, if $G$ is divisible, $k[t^G]$ is real closed if $k$ is real closed. This is because the adjunction of a square root of $-1$ would make $k[t^G]$ algebraically closed, since this gives the same result as constructing the Hahn series over the algebraically closed field $k[\sqrt{-1}]$.

The multiplicative valuation $v(f)$ of an element $f\in k[t^G]$ is the least $x \in G$ for which $f(x) \neq 0$. This yields a valuation ring, such a valuation ring determines and is determined by a valuation on a field. The field $k[t^G]$ is a complete ultrametric space (and indeed a spherically complete field?) with respect to this valuation, but unlike for formal power series the formal series $\sum_{x\in G} f(x) t^x$ do not actually “converge” in this metric.

Examples

• If $G=\mathbb{R}$, then we have Hahn series such as

  $$t^1 + t^{3/2} + t^{5/3} + t^{7/4} + \cdots$$

• Via Conway normal forms, the ring of Hahn series with $k=\mathbb{R}$ and with value group the surreal numbers is isomorphic to the surreal numbers themselves. In other words, the surreals are a fixed point of the “Hahn series with real coefficients” functor on the category of abelian groups. However, the surreals are not the initial fixed point of this functor, since there are surreals that appear as exponents in their own Conway normal form — for instance, the ∞-numbers.

• Well-based transseries can be constructed by iterating the Hahn series construction.

• If $G=\mathbb{Z}$, then a Hahn series is precisely a formal Laurent series?, i.e. $k[t^{\mathbb{Z}}] = k((t))$.

• If $G=\mathbb{Q}$, then $k[t^{\mathbb{Q}}]$ properly contains the Levi-Civita field, as the subclass of those Hahn series whose support is finite below any fixed rational.

• More generally, any Hahn series field $k[t^G]$ contains the Novikov field of $k$ with value group $G$.

Related pages

• formal power series
• formal Laurent series?
• Puiseux series
• Novikov series
• Ribenboim power series
• finiteness space