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

Authors: Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven

Keywords: H-fields, fields of transseries

A.M.S. subject classification: 03C64, 16W60, 26A12

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