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Transseries as germs of surreal functions

Alessandro Berarducci, Vincenzo Mantova

Abstract

We show that Écalle's transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper.

Transseries as germs of surreal functions

Abstract

We show that Écalle's transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper.

Paper Structure

This paper contains 34 sections, 71 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hahn fields
  4. Surreal numbers
  5. Summability
  6. Hahn fields embedded in $\mathbf{No}$
  7. Surreal analytic functions
  8. Products and powers of summable families
  9. Sums of power series
  10. Composition of power series
  11. Transseries
  12. Log-atomic numbers
  13. Omega-series, LE-series, EL-series
  14. Isomorphism with classical LE-series
  15. Adding more log-atomic numbers
  16. ...and 19 more sections

Key Result

Proposition 2.16

Let $\Gamma$ be a small multiplicative subgroup of $\mathfrak{M}=e^{\mathbb{J}}$ and $R$ be a truncation closed subfield of $\mathbf{No}$. If $R<\Gamma^{>1}$, there is a unique field embedding $R((\Gamma))\to\mathbf{No}$ sending $r\mathfrak{m}$ (as an element of $R((\Gamma))$) to $r\mathfrak{m}$ (as

Figures (1)

  • Figure 5.1: An example of tree with root $R(T)=re^{\gamma}$, where $se^{\lambda}$ is a term of $\gamma$, $\lambda\in\Delta$, $t_{0},\dots,t_{m-1}$ are terms of $c_{0}(\lambda)$, and the contribution $\overline{c}(T)$ of $T$ is $\mkern-135mu{ \overline{c}(T)}={ re^{c(\gamma)^{\uparrow=}}\frac{1}{n!}\overline{c}(\tau(0))\overline{c}(\tau(1))\ldots}={ re^{c(\gamma)^{\uparrow=}}\frac{1}{n!}se^{c_{0}(\lambda)^{\uparrow=}}\frac{1}{m!}\overline{c}(\sigma(0))\ldots\overline{c}(\sigma(m-1))\overline{c}(\tau(1))\ldots}={ re^{c(\gamma)^{\uparrow=}}\frac{1}{n!}se^{c_{0}(\lambda)^{\uparrow=}}\frac{1}{m!}t_{0}\dots t_{m-1}\overline{c}(\tau(1))\ldots}$

Theorems & Definitions (192)

  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.6
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • Example 2.13
  • ...and 182 more