Homogeneous Universal H-fields

TL;DR

This paper characterizes a unique, homogeneous, universal derivation on Conway's surreal numbers, making the structure a canonical model of a specific class of ordered differential fields with real constants.

Contribution

It establishes the uniqueness and universality of a particular derivation on surreal numbers, providing a canonical model for certain $H$-fields with small derivation.

Findings

The derivation on $ extbf{No}$ is unique up to isomorphism.

The structure is absolutely homogeneous and universal.

It models the theory of $H$-fields with small derivation and constant field $ r$.

Abstract

We consider derivations $\partial$ on Conway's field $\mathbf{No}$ of surreal numbers such that the ordered differential field $(\mathbf{No},\partial)$ has constant field $\mathbb{R}$ and is a model of the model companion of the theory of $H$-fields with small derivation. We show that this determines $(\mathbf{No},\partial)$ uniquely up to isomorphism, and that this structure is absolutely homogeneous universal for models of this theory with constant field $\mathbb{R}$.