Monotonicity and a Taylor approximation theorem for transseries

TL;DR

This paper proves monotonicity and a Taylor approximation theorem for transseries, including omega-series and LE-series, with implications for their functional properties.

Contribution

It establishes monotonicity of composition in transseries and introduces a Taylor approximation theorem with maximal radius of validity.

Findings

Composition of omega-series is monotonic in its second argument.

Omega-series and LE-series have the intermediate value property.

A Taylor approximation theorem with maximal radius of validity is proved.

Abstract

We show that the composition of omega-series by surreal numbers, or more generally by elements of any confluent field of transseries, is monotonic in its second argument. In particular, omega-series and LE-series interpreted as functions have the intermediate value property. We also deduce a Taylor approximation theorem for omega-series with maximal radius of validity.