Surreal numbers, derivations and transseries

TL;DR

This paper proves that Conway's surreal numbers, with an exponential and differential structure, form a Liouville closed field of transseries, confirming longstanding conjectures and establishing their rich algebraic and differential properties.

Contribution

It provides a complete positive solution to conjectures about surreal numbers being a field of transseries with a Hardy-type derivation.

Findings

Surreal numbers can be equipped with a compatible exponential and differential structure.

The differential structure makes surreal numbers Liouville closed, with a surjective derivation.

Surreal numbers can be fully described as a field of transseries.

Abstract

Several authors have conjectured that Conway's field of surreal numbers, equipped with the exponential function of Kruskal and Gonshor, can be described as a field of transseries and admits a compatible differential structure of Hardy-type. In this paper we give a complete positive solution to both problems. We also show that with this new differential structure, the surreal numbers are Liouville closed, namely the derivation is surjective.