ASCHENBRENNER, Matthias - Analyse Numérique et E.D.P., University of California at Berkeley, Berkeley, CA 94720, USA.
VAN DER DRIES, Lou - Analyse Numérique et E.D.P., University of Illinois at Urbana-Champaign, Urbana, IL 61810, USA.
VAN DER HOEVEN, Joris - Analyse Harmonique, Université Paris-Sud, Bât. 425, 91405 Orsay cedex

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.

Contact : Joris.VanDerHoeven@math.u-psud.fr