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Hahn series

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 GG be an ordered abelian group, and kk a field.

Definition

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

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

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

Properties

Theorem

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

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

The multiplicative valuation v(f)v(f) of an element fk[t G]f\in k[t^G] is the least xG opx \in G^{op} for which f(x)0f(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]k[t^G] is complete with respect to this valuation, but unlike for formal power series the formal series xGf(x)t x\sum_{x\in G} f(x) t^x do not actually “converge” in this metric.

Examples

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

    t 1+t 3/2+t 4/3+t 5/4+ t^1 + t^{3/2} + t^{4/3} + t^{5/4} + \cdots
  • Via Conway normal forms, the ring of Hahn series with k=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 ε-numbers.

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

Revised on August 26, 2016 17:10:25 by Mike Shulman (76.167.222.204)