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

e-mail: , ,
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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