Real Algebraic and Analytic Geometry |

Differentially Algebraic Gaps.

e-mail: ,
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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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