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Monotonicity and a Taylor approximation theorem for transseries

Vincenzo Mantova

TL;DR

The paper addresses the problem of monotonicity for left composition by omega-series and related transseries on positive infinite elements of a confluent transseries field. It develops a two-stage approach: first proving strict monotonicity for purely infinite series via induction on the exponential rank and a weak mean-value property, then extending to general omega-series and establishing an intermediate-value property for compositions. It also proves a Taylor-type approximation with maximal radius of validity for omega-series, clarifying error terms and radius demonstrations. These results advance the composition calculus in transseries and surreal-number–based frameworks, with potential implications for asymptotic analysis, Hardy fields, and hyperserial extensions.

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 on positive infinite omega-series, or respectively LE-series, have the intermediate value property. We also deduce a Taylor approximation theorem for omega-series with maximal radius of validity.

Monotonicity and a Taylor approximation theorem for transseries

TL;DR

The paper addresses the problem of monotonicity for left composition by omega-series and related transseries on positive infinite elements of a confluent transseries field. It develops a two-stage approach: first proving strict monotonicity for purely infinite series via induction on the exponential rank and a weak mean-value property, then extending to general omega-series and establishing an intermediate-value property for compositions. It also proves a Taylor-type approximation with maximal radius of validity for omega-series, clarifying error terms and radius demonstrations. These results advance the composition calculus in transseries and surreal-number–based frameworks, with potential implications for asymptotic analysis, Hardy fields, and hyperserial extensions.

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 on positive infinite omega-series, or respectively LE-series, have the intermediate value property. We also deduce a Taylor approximation theorem for omega-series with maximal radius of validity.
Paper Structure (5 sections, 11 theorems, 41 equations)

This paper contains 5 sections, 11 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Some preliminaries
  3. Monotonicity for purely infinite series
  4. Proof of monotonicity
  5. Taylor's approximation

Key Result

Theorem 1

For all $f \in \mathbb{R}\langle\!\langle T\rangle\!\rangle$, the function $x \mapsto f \circ x$ for $x \in \mathbb{U}^{>\mathbb{R}}$ is strictly increasing if $f' > 0$, strictly decreasing if $f' < 0$, and constant if $f' = 0$.

Theorems & Definitions (24)

  • Theorem 1: Monotonicity
  • Corollary 2
  • Theorem 3: Taylor approximation for $\mathbb{R}\langle\!\langle T\rangle\!\rangle$
  • Remark 2.1
  • Definition 2.3
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Lemma 4.1
  • ...and 14 more