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MODEL THEORY OF DIFFERENTIAL-HENSELIAN PRE-H-FIELDS

Part of: Connections with logic Topological fields Differential and difference algebra Model theory

Published online by Cambridge University Press:  08 April 2025

NIGEL PYNN-COATES Open the ORCID record for NIGEL PYNN-COATES [Opens in a new window]
NIGEL PYNN-COATES*
Affiliation:
KURT GÖDEL RESEARCH CENTER INSTITUTE OF MATHEMATICS UNIVERSITY OF VIENNA WIEN AUSTRIA
*
E-mail: nigel.pynn-coates@univie.ac.at
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Abstract

Pre-H-fields are ordered valued differential fields satisfying some basic axioms coming from transseries and Hardy fields. We study pre-H-fields that are differential-Hensel–Liouville closed, that is, differential-henselian, real closed, and closed under exponential integration, establishing an Ax–Kochen/Ershov theorem for such structures: the theory of a differential-Hensel–Liouville closed pre-H-field is determined by the theory of its ordered differential residue field; this result fails if the assumption of closure under exponential integration is dropped. In a two-sorted setting with one sort for a differential-Hensel–Liouville closed pre-H-field and one sort for its ordered differential residue field, we eliminate quantifiers from the pre-H-field sort, from which we deduce that the ordered differential residue field is purely stably embedded and if it has NIP, then so does the two-sorted structure. Similarly, the one-sorted theory of differential-Hensel–Liouville closed pre-H-fields with closed ordered differential residue field has quantifier elimination, is the model completion of the theory of pre-H-fields with gap $0$, and is complete, distal, and locally o-minimal.

Keywords

valued fieldsdifferential fieldsordered fieldsmodel theory

MSC classification

Primary: 03C64: Model theory of ordered structures; o-minimality 12H05: Differential algebra
Secondary: 03C10: Quantifier elimination, model completeness and related topics 03C60: Model-theoretic algebra 12J10: Valued fields 12J15: Ordered fields 12L12: Model theory

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Type
Article
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The Journal of Symbolic Logic , First View , pp. 1 - 40
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

REFERENCES

Aschenbrenner, M., Some remarks about asymptotic couples , Valuation Theory and its Applications (Saskatoon, SK, 1999) (F.-V. Kuhlmann, S. Kuhlmann, and M. Marshall, editors), vol. 2, Fields Institute Communications, 33, American Mathematical Society, Providence, 2003, pp. 7–18.Google Scholar
Aschenbrenner, M., Chernikov, A., Gehret, A., and Ziegler, M., Distality in valued fields and related structures . Transactions of the American Mathematical Society, vol. 375 (2022), no. 7, pp. 4641–4710.Google Scholar
Aschenbrenner, M. and van den Dries, L., Closed asymptotic couples . Journal of Algebra, vol. 225 (2000), no. 1, pp. 309–358. https://doi.org/10.1006/jabr.1999.8128.Google Scholar
Aschenbrenner, M. and van den Dries, L.,  $H$ -fields and their Liouville extensions . Mathematische Zeitschrift, vol. 242 (2002), no. 3, pp. 543–588. https://doi.org/10.1007/s002090000358.Google Scholar
Aschenbrenner, M., van den Dries, L., and van der Hoeven, J., Asymptotic Differential Algebra and Model Theory of Transseries, vol. 195, Annals of Mathematics Studies, Princeton University Press, Princeton, 2017, pp. xxi+849. https://doi.org/10.1515/9781400885411.Google Scholar
Aschenbrenner, M., van den Dries, L., and van der Hoeven, J., Revisiting closed asymptotic couples . Proceedings of the Edinburgh Mathematical Society (2), vol. 65 (2022), no. 2, pp. 530–555. https://doi.org/10.1017/S0013091522000219.Google Scholar
Ax, J. and Kochen, S., Diophantine problems over local fields: III: Decidable fields . Annals of Mathematics (2), vol. 83 (1966), pp. 437–456. https://doi.org/10.2307/1970476.Google Scholar
Bélair, L. and Bousquet, M., Types dans les corps valués . Comptes Rendus de l’Académie des Sciences, vol. 323 (1996), no. 8, pp. 841–844.Google Scholar
Boshernitzan, M., Hardy fields and existence of transexponential functions . Aequationes Mathematicae, vol. 30 (1986), nos. 2–3, pp. 258–280. https://doi.org/10.1007/BF02189932.Google Scholar
Delon, F., Types sur $C((X))$ , Study Group on Stable Theories (Bruno Poizat) Second year: 1978/79 (French), Secrétariat Math., Paris, 1981. Exp. No. 5, 29.Google Scholar
Ershov, Y. L., On the elementary theory of maximal normed fields . Doklady Akademii Nauk SSSR, vol. 165 (1965), pp. 21–23.Google Scholar
Hakobyan, T., An Ax-Kochen-Ershov theorem for monotone differential-Henselian fields . The Journal of Symbolic Logic, vol. 83 (2018), no. 2, pp. 804–816. https://doi.org/10.1017/jsl.2017.64.Google Scholar
Marker, D., Model Theory: An Introduction, vol. 217, Graduate Texts in Mathematics, Springer-Verlag, New York, 2002, pp. viii+342.Google Scholar
Pynn-Coates, N., Differential-henselianity and maximality of asymptotic valued differential fields . Pacific Journal of Mathematics, vol. 308 (2020), no. 1, pp. 161–205. https://doi.org/10.2140/pjm.2020.308.Google Scholar
Pynn-Coates, N., Tame pairs of transseries fields, Preprint. 2024. https://arxiv.org/abs/2408.07033v1.Google Scholar
Pynn-Coates, N., “Dimension and topology in transserial tame pairs”, 2025, preprint at url: https://arxiv.org/abs/2508.16415.Google Scholar
Pynn-Coates, N., “On asymptotic valued differential fields with small derivation”, PhD thesis, University of Illinois at Urbana-Champaign, 2020.Google Scholar
Rosenlicht, M., On the value group of a differential valuation . American Journal of Mathematics, vol. 101 (1979), no. 1, pp. 258–266. https://doi.org/10.2307/2373949.Google Scholar
Rosenlicht, M., Differential valuations . Pacific Journal of Mathematics, vol. 86 (1980), no. 1, pp. 301–319. url: https://projecteuclid.org/euclid.pjm/1102780625.Google Scholar
Rosenlicht, M., On the value group of a differential valuation. II . American Journal of Mathematics, vol. 103 (1981), no. 5, pp. 977–996. https://doi.org/10.2307/2374255.Google Scholar
Scanlon, T., A model complete theory of valued $D$ -fields . Journal of Symbolic Logic, vol. 65 (2000), no. 4, pp. 1758–1784. https://doi.org/10.2307/2695074.Google Scholar
Singer, M. F., The model theory of ordered differential fields . Journal of Symbolic Logic, vol. 43 (1978), no. 1, pp. 82–91. https://doi.org/10.2307/2271951.Google Scholar
van den Dries, L., Macintyre, A., and Marker, D., Logarithmic-exponential power series . Journal of the London Mathematical Society (2), vol. 56 (1997), no. 3, pp. 417–434. https://doi.org/10.1112/S0024610797005437.Google Scholar