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Closed asymptotic couples. (English) Zbl 0974.12015

The authors investigate asymptotic couples of Hardy type in the sense of M. Rosenlicht [Am. J. Math. 103, 977-996 (1981; Zbl 0474.12020)]. The asymptotic couples with scalar multiplication associated to Hardy fields \(F\) belong to a certain elementary class, the class of H-couples. If \(F\) is a maximal Hardy field, its associated H-couple is closed (The authors give precise definitions in Section 1 of the paper.) The main result is that closed H-couples have an elimination theory. L. van den Dries, A. J. Macintyre and D. E. Marker in one of the most influential papers in model theory of the 1990s [Ann. Math. (2) 140, No. 1, 183-205 (1994; Zbl 0837.12006)] established o-minimality of the real field augmented by exponentiation and all restricted analytic functions using the theory of Hardy fields and some non-trivial facts on quantifier elimination. The paper under review can be seen as a very interesting step towards generalization of the o-minimality results mentioned above.
Reviewer: Z.Hajto (Kraków)

MSC:

12J10 Valued fields
03C60 Model-theoretic algebra
12L12 Model theory of fields
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References:

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