Real Algebraic and Analytic Geometry

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63. Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven:
Differentially Algebraic Gaps.

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Submission: 2003, October 19.

$H$-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 \cite{AvdD99}. The key is a combinatorial fact about the support of transseries obtained from iterated logarithms by algebraic operations, integration, and exponentiation.

Mathematics Subject Classification (2000): 03C64, 16W60, 26A12.

Keywords and Phrases: $H$-fields, fields of transseries.

Full text, 30p.: dvi 158k, ps.gz 264k, pdf 345k.

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