Real Algebraic and Analytic Geometry Preprint Server

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homepages: http://www.math.uic.edu/~maschenb/, http://www.math.uiuc.edu/People/vddries.html, http://www.math.u-psud.fr/~vdhoeven/

Submission: 2003, October 19.

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 \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.

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