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LIOUVILLE CLOSED H-FIELDS MATTHIAS ASCHENBRENNER AND LOU VAN DEN DRIES
 

Summary: LIOUVILLE CLOSED H-FIELDS
MATTHIAS ASCHENBRENNER AND LOU VAN DEN DRIES
Abstract. H-fields are fields with an ordering and a derivation sub-
ject to some compatibilities. (Hardy fields extending R and fields of
transseries over R are H-fields.) We prove basic facts about the loca-
tion 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 ob-
taining a good model theory of H-fields, including an undecidability
result. We finish with open questions that motivate our work.
Contents
Introduction 2
1. Asymptotic Relations and Exponentiation 3
2. Nonexistence of Small Infinite Zeros 7
3. Nonexistence of Large Infinite Zeros 9
4. Intermediate Value Property
for First-Order Differential Polynomials 13
5. The Valuation of Higher Derivatives 17
6. Simple Zeros of Differential Polynomials 19

  

Source: Aschenbrenner, Matthias - Department of Mathematics, University of California at Los Angeles

 

Collections: Mathematics