Real Algebraic and Analytic Geometry
homepages: http://www.math.uic.edu/~maschenb/, http://www.math.uiuc.edu/People/vddries.html
Submission: 2002, November 7.
$H$-fields are fields with an ordering and a derivation subject to some compatibilities. (Hardy fields extending $\R$ and fields of transseries over $\R$ are $H$-fields.) We prove basic facts about the location of zeros of differential polynomials in Liouville closed $H$-fields, and study various constructions in the category of $H$-fields: closure under powers, constant field extension, completion, and building $H$-fields with prescribed constant field and $H$-couple. We indicate difficulties in obtaining a good model theory of $H$-fields, including an undecidability result. We finish with open questions that motivate our work.
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