Differentially algebraic gaps

Matthias Aschenbrenner, Lou Van Den Dries, Joris Van Der Hoeven

Research output: Contribution to journalArticle

Abstract

H-fields are ordered differential fields that capture some basic properties of Hardy fields and fields of transseries. Each H-field is equipped with a convex valuation, and solving first-order linear differential equations in H-field extensions is strongly affected by the presence of a "gap" in the value group. We construct a real closed H-field that solves every first-order linear differential equation, and that has a differentially algebraic H-field extension with a gap. This answers a question raised in [1]. The key is a combinatorial fact about the support of transseries obtained from iterated logarithms by algebraic operations, integration, and exponentiation.

Original languageEnglish (US)
Pages (from-to)247-280
Number of pages34
JournalSelecta Mathematica, New Series
Volume11
Issue number2
DOIs
StatePublished - Jun 1 2005

Fingerprint

differential equations
logarithms
Field extension
First order differential equation
Linear differential equation
Exponentiation
Logarithm
Valuation
Closed

Keywords

  • Fields of transseries
  • H-fields

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)

Cite this

Differentially algebraic gaps. / Aschenbrenner, Matthias; Van Den Dries, Lou; Van Der Hoeven, Joris.

In: Selecta Mathematica, New Series, Vol. 11, No. 2, 01.06.2005, p. 247-280.

Research output: Contribution to journalArticle

Aschenbrenner, Matthias ; Van Den Dries, Lou ; Van Der Hoeven, Joris. / Differentially algebraic gaps. In: Selecta Mathematica, New Series. 2005 ; Vol. 11, No. 2. pp. 247-280.
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