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The differential rank of a differential-valued field

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Salma Kuhlmann
Salma Kuhlmann
Salma Kuhlmann
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Gabriel Lehéricy
Gabriel Lehéricy
Gabriel Lehéricy
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Abstract

We develop a notion of (principal) differential rank for differential-valued fields, in analog of the exponential rank developed in Kuhlmann (The Fields Institute Monograph Series 12, 2000) and of the difference rank developed in Kuhlmann (Groups, Modules and Model Theory—Surveys and Recent Developments in Memory of Rdiger Gbel, pp 399–414, 2017). We give several characterizations of this rank. We then give a method to define a derivation on a field of generalized power series and use this method to show that any totally ordered set can be realized as the principal differential rank of a H-field.
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Mathematische Zeitschrift (2019) 292:1017–1049
https://doi.org/10.1007/s00209-018-2132-z
Mathematische Zeitschrift
The differential rank of a differential-valued field
Salma Kuhlmann1·Gabriel Lehéricy1
Received: 14 August 2017 / Accepted: 26 June 2018 / Published online: 7 August 2018
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract
We develop a notion of (principal) differential rank for differential-valued fields, in analog
of the exponential rank developed in Kuhlmann (The Fields Institute Monograph Series
12, 2000) and of the difference rank developed in Kuhlmann (Groups, Modules and Model
Theory—Surveys and Recent Developments in Memory of Rdiger Gbel, pp 399–414, 2017).
We give several characterizations of this rank. We then give a method to define a derivation
on a field of generalized power series and use this method to show that any totally ordered
set can be realized as the principal differential rank of a H-field.
Keywords Valuation ·Differential field ·Differential-valued field ·H-field ·Asymptotic
couple ·Generalized power series
Mathematics Subject Classification 12H05 ·12J10 ·12J15 ·13F25 ·16W60
1 Introduction
The rank of a valued field (i.e the order-type of the set of coarsenings of the given valuation,
ordered by inclusion, see [6,19]) is an important invariant. It has three equivalent charac-
terizations: one at the level of the valued field (K,v) itself, another one at the level of the
value group G:= v(K×)and a third one at the level of the value chain := vG(G=0)of
the value group. Recently, notions of ranks have been developed for valued fields endowed
with an operator; examples of this are the exponential rank of an ordered exponential field
(see [12, Chapter 3, Section 2]) and the difference rank of a valued difference field (see [16,
Section 4]). In this paper, we are interested in pre-differential-valued fields as introduced by
Aschenbrenner and v.d.Dries in [3], i.e valued fields endowed with a derivation which is in
some sense compatible with the valuation (see Sect. 5below for the precise definition). We
will pay a special attention to the class of H-fields (see Definition 5.1) introduced by Aschen-
brenner and v.d.Dries in [3]and[4]. H-fields are a generalization of Hardy fields (introduced
by Hardy in [9]). H-fields are particularly interesting because of the central role they play in
the theory of transseries and in the model-theoretic study of Hardy fields (see [5]). A sub-
stantial part of our work relies on the theory of asymptotic couples. Asymptotic couples were
BGabriel Lehéricy
gabriel.lehericy@uni-konstanz.de
1Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Constance, Germany
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... The q.o contains both the information about the order ≤ of K and some information about the σ-rank of K. Note that we can do a similar construction with H-fields if we replace σ by the q.o φ defined in Section 3.2 of [14]. ...
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