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.
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Aschenbrenner, M., Dries, L.v.d. & Hoeven, J.v.d. Differentially algebraic gaps. Sel. math., New ser. 11, 247 (2005). https://doi.org/10.1007/s00029-005-0010-0
DOI: https://doi.org/10.1007/s00029-005-0010-0