Differentially algebraic gaps

• Matthias Aschenbrenner ^[3] ; Lou van den Dries ^[1] ; Joris van der Hoeven ^[2]
  1. [1] University of Illinois at Urbana Champaign

     University of Illinois at Urbana Champaign

     Township of Cunningham, Estados Unidos

     
  2. [2] University of Paris-Sud

     University of Paris-Sud

     Arrondissement de Palaiseau, Francia

     
  3. [3] Department of Mathematics, University of California at Berkeley, USA

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 11, Nº. 2, 2005, págs. 247-280
• Idioma: inglés
• DOI: 10.1007/s00029-005-0010-0
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• Resumen

  • 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.