Distality for the asymptotic couple of the field of logarithmic transseries. (English) Zbl 1484.03068

Summary: We show that the theory \(T_{\log}\) of the asymptotic couple of the field of logarithmic transseries is distal. As distal theories are NIP (have the non-independence property), this provides a new proof that \(T_{\log}\) is NIP.


03C64 Model theory of ordered structures; o-minimality
03C45 Classification theory, stability, and related concepts in model theory
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
12L12 Model theory of fields
Full Text: DOI arXiv Euclid


[1] Aschenbrenner, M., L. van den Dries, and J. van der Hoeven, Asymptotic Differential Algebra and Model Theory of Transseries, vol. 195 of Annals of Mathematics Studies, Princeton University Press, Princeton, 2017. · Zbl 1430.12002
[2] Chernikov, A., D. Galvin, and S. Starchenko, “Cutting lemma and Zarankiewicz’s problem in distal structures,” preprint, arXiv:1612.00908v1 [math.LO]. · Zbl 1485.03092
[3] Chernikov, A., and P. Simon, “Externally definable sets and dependent pairs, II,” Transactions of the American Mathematical Society, vol. 367 (2015), pp. 5217-35. · Zbl 1388.03035
[4] Chernikov, A., and S. Starchenko, “Regularity lemma for distal structures,” Journal of the European Mathematical Society (JEMS), vol. 20 (2018), pp. 2437-66. · Zbl 1459.03041
[5] Dolich, A., and J. Goodrick, “Strong theories of ordered Abelian groups,” Fundamenta Mathematicae, vol. 236 (2017), pp. 269-96. · Zbl 1420.03065
[6] Gehret, A., “The asymptotic couple of the field of logarithmic transseries,” Journal of Algebra, vol. 470 (2017), pp. 1-36. · Zbl 1423.03133
[7] Gehret, A., “NIP for the asymptotic couple of the field of logarithmic transseries,” Journal of Symbolic Logic, vol. 82 (2017), pp. 35-61. · Zbl 1419.03035
[8] Gehret, A., “A tale of two Liouville closures,” Pacific Journal of Mathematics, vol. 290 (2017), pp. 41-76. · Zbl 1387.12004
[9] Gehret, A., “Towards a model theory of logarithmic transseries,” Ph.D. dissertation, University of Illinois at Urbana-Champaign, Illinois, 2017. · Zbl 1423.03133
[10] Günaydin, A., and P. Hieronymi, “Dependent pairs,” Journal of Symbolic Logic, vol. 76 (2011), pp. 377-90. · Zbl 1220.03031
[11] Hieronymi, P., and T. Nell, “Distal and non-distal pairs,” Journal of Symbolic Logic, vol. 82 (2017), pp. 375-83. · Zbl 1419.03036
[12] Rosenlicht, M., “On the value group of a differential valuation, II,” American Journal of Mathematics, vol. 103 (1981), pp. 977-96. · Zbl 0474.12020
[13] Simon, P., “Distal and non-distal NIP theories,” Annals of Pure and Applied Logic, vol. 164 (2013), pp. 294-318. · Zbl 1269.03037
[14] Simon, P., A Guide to NIP Theories, vol. 44 of Lecture Notes in Logic, Cambridge University Press, Cambridge, 2015.
[15] Tent, K. · Zbl 1245.03002
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