Skip to main content
Log in

Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven: “Asymptotic Differential Algebra and Model Theory of Transseries”

Princeton University Press, 2017, 880 pp.

  • Book Review
  • Published:
Jahresbericht der Deutschen Mathematiker-Vereinigung Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Artin, E.: Über die Zerlegung definiter Funktionen in Quadrate. Abh. Math. Semin. Univ. Hamb. 5(1), 100–115 (1927)

    Article  Google Scholar 

  2. Artin, E., Schreier, O.: Eine Kennzeichnung der reell abgeschlossenen Körper. Abh. Math. Semin. Univ. Hamb. 5(1), 225–231 (1927)

    Article  Google Scholar 

  3. Ax, J.: On Schanuel’s conjectures. Ann. Math. (2) 93, 252–268 (1971)

    Article  MathSciNet  Google Scholar 

  4. Berarducci, A., Ehrlich, P., Kuhlmann, S.: Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations. Oberwolfach Rep. 13(4), 3313–3372 (2016)

    Article  MathSciNet  Google Scholar 

  5. Berarducci, A., Mantova, V.: Surreal numbers, derivations and transseries. J. Eur. Math. Soc. 20(2), 339–390 (2018)

    Article  MathSciNet  Google Scholar 

  6. Conway, J.H.: On Numbers and Games, 2nd edn. A K Peters, Ltd., Natick (2001)

    MATH  Google Scholar 

  7. Dahn, B.I., Göring, P.: Notes on exponential-logarithmic terms. Fundam. Math. 127(1), 45–50 (1987)

    Article  MathSciNet  Google Scholar 

  8. Écalle, J.: Six lectures on transseries, analysable functions and the constructive proof of Dulac’s conjecture. In: Bifurcations and Periodic Orbits of Vector Fields, Montreal, PQ, 1992. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 408, pp. 75–184. Kluwer Acad. Publ., Dordrecht (1993)

    Chapter  Google Scholar 

  9. Hahn, H.: Gesammelte Abhandlungen/Collected Works. Band 1/Vol. 1. Springer, Vienna (1995). With biographical sketches by Karl Popper and by L. Schmetterer and K. Sigmund, and commentaries on Hahn’s work by H. Heuser, H. Sagan and L. Fuchs, Edited by Schmetterer and Sigmund and with a foreword by Popper

    Google Scholar 

  10. Hardy, G.H.: Orders of Infinity. The Infinitärcalcül of Paul du Bois-Reymond. Cambridge Tracts in Mathematics and Mathematical Physics, vol. 12. Hafner Publishing Co., New York (1971). Reprint of the 1910 edition

    Google Scholar 

  11. Henrion, D., Infusino, M., Kuhlmann, S., Vinnikov, V.: Real Algebraic Geometry with a View Toward Moment Problems and Optimization. Oberwolfach Rep. 14(1), 771–862 (2017)

    Article  MathSciNet  Google Scholar 

  12. Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. 8(10), 437–479 (1902)

    Article  MathSciNet  Google Scholar 

  13. Kaplansky, I.: Maximal fields with valuations. Duke Math. J. 9, 303–321 (1942)

    Article  MathSciNet  Google Scholar 

  14. Kaplansky, I.: An Introduction to Differential Algebra, 2nd edn. Actualités Scientifiques et Industrielles, vol. 1251. Hermann, Paris (1976). Publications de l’Institut de Mathématique de l’Université de Nancago, No. V

    MATH  Google Scholar 

  15. Krull, W.: Gesammelte Abhandlungen/Collected Papers. Vol. 1, 2. Walter de Gruyter & Co., Berlin (1999). With biographical contributions by H. Schöneborn, H.-J. Nastold, J. Neukirch and Paulo Ribenboim, Edited and with a preface by Ribenboim

    MATH  Google Scholar 

  16. Kuhlmann, F.-V., Kuhlmann, S., Shelah, S.: Exponentiation in power series fields. Proc. Am. Math. Soc. 125(11), 3177–3183 (1997)

    Article  MathSciNet  Google Scholar 

  17. Kuhlmann, S.: Ordered Exponential Fields. Fields Institute Monographs, vol. 12. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  18. Kuhlmann, S., Matusinski, M.: Hardy type derivations on fields of exponential logarithmic series. J. Algebra 345, 171–189 (2011)

    Article  MathSciNet  Google Scholar 

  19. Kuhlmann, S., Matusinski, M.: Hardy type derivations on generalised series fields. J. Algebra 351, 185–203 (2012)

    Article  MathSciNet  Google Scholar 

  20. Kuhlmann, S., Matusinski, M.: The exponential-logarithmic equivalence classes of surreal numbers. Order 32(1), 53–68 (2015)

    Article  MathSciNet  Google Scholar 

  21. Kuhlmann, S., Matusinski, M., Shkop, A.C.: A note on Schanuel’s conjectures for exponential logarithmic power series fields. Arch. Math. (Basel) 100(5), 431–436 (2013)

    Article  MathSciNet  Google Scholar 

  22. Kuhlmann, S., Shelah, S.: \(\kappa \)-bounded exponential-logarithmic power series fields. Ann. Pure Appl. Logic 136(3), 284–296 (2005)

    Article  MathSciNet  Google Scholar 

  23. Bernard Lasserre, J.: Moments, Positive Polynomials and Their Applications. Imperial College Press Optimization Series, vol. 1. Imperial College Press, London (2010)

    MATH  Google Scholar 

  24. Mac Lane, S.: The universality of formal power series fields. Bull. Am. Math. Soc. 45, 888–890 (1939)

    Article  MathSciNet  Google Scholar 

  25. Vincenzo, M., Matusinski, M.: Surreal numbers with derivation, Hardy fields and transseries: a survey. Ordered algebraic structures and related topics. In: Contemp. Math., vol. 697, pp. 265–290. Amer. Math. Soc., Providence (2017)

    Google Scholar 

  26. Matusinski, M.: On generalized series fields and exponential-logarithmic series fields with derivations. In: Valuation Theory in Interaction. EMS Ser. Congr. Rep., pp. 350–372. Eur. Math. Soc., Zürich (2014)

    Google Scholar 

  27. Robinson, A.: On the real closure of a Hardy field. In: Theory of Sets and Topology, pp. 427–433 (1972)

    Google Scholar 

  28. Robinson, A.: Ordered differential fields. J. Comb. Theory, Ser. A 14, 324–333 (1973)

    Article  MathSciNet  Google Scholar 

  29. Rosenlicht, M.: An analogue of l’Hospital’s rule. Proc. Am. Math. Soc. 37, 369–373 (1973)

    MathSciNet  MATH  Google Scholar 

  30. Rosenlicht, M.: The nonminimality of the differential closure. Pac. J. Math. 52, 529–537 (1974)

    Article  MathSciNet  Google Scholar 

  31. Rosenlicht, M.: Differential extension fields of exponential type. Pac. J. Math. 57(1), 289–300 (1975)

    Article  MathSciNet  Google Scholar 

  32. Rosenlicht, M.: On Liouville’s theory of elementary functions. Pac. J. Math. 65(2), 485–492 (1976)

    Article  MathSciNet  Google Scholar 

  33. Rosenlicht, M.: On the value group of a differential valuation. Am. J. Math. 101(1), 258–266 (1979)

    Article  MathSciNet  Google Scholar 

  34. Rosenlicht, M.: Differential valuations. Pac. J. Math. 86(1), 301–319 (1980)

    Article  MathSciNet  Google Scholar 

  35. Rosenlicht, M.: On the value group of a differential valuation. II. Am. J. Math. 103(5), 977–996 (1981)

    Article  MathSciNet  Google Scholar 

  36. Rosenlicht, M.: Hardy fields. J. Math. Anal. Appl. 93(2), 297–311 (1983)

    Article  MathSciNet  Google Scholar 

  37. Rosenlicht, M.: The rank of a Hardy field. Trans. Am. Math. Soc. 280(2), 659–671 (1983)

    Article  MathSciNet  Google Scholar 

  38. Rosenlicht, M.: Rank change on adjoining real powers to Hardy fields. Trans. Am. Math. Soc. 284(2), 829–836 (1984)

    Article  MathSciNet  Google Scholar 

  39. Rosenlicht, M.: Growth properties of functions in Hardy fields. Trans. Am. Math. Soc. 299(1), 261–272 (1987)

    Article  MathSciNet  Google Scholar 

  40. Rosenlicht, M.: Asymptotic solutions of \(Y''=F(x)Y\). J. Math. Anal. Appl. 189(3), 640–650 (1995)

    Article  MathSciNet  Google Scholar 

  41. Rosenlicht, M., Singer, M.: On elementary, generalized elementary, and Liouvillian extension fields. In: Contributions to Algebra, pp. 329–342 (1977)

    Chapter  Google Scholar 

  42. Scanlon, T.: A model complete theory of valued \(D\)-fields. J. Symb. Log. 65(4), 1758–1784 (2000)

    Article  MathSciNet  Google Scholar 

  43. Scanlon, T.: Differentially valued fields are not differentially closed. In: Model Theory with Applications to Algebra and Analysis. Vol. 1. London Math. Soc. Lecture Note Ser., vol. 349, pp. 111–115. Cambridge Univ. Press, Cambridge (2008)

    Chapter  Google Scholar 

  44. Seidenberg, A.: A new decision method for elementary algebra. Ann. Math. (2) 60, 365–374 (1954)

    Article  MathSciNet  Google Scholar 

  45. Singer, M.F.: A class of differential fields with minimal differential closures. Proc. Am. Math. Soc. 69(2), 319–322 (1978)

    Article  MathSciNet  Google Scholar 

  46. Singer, M.F.: The model theory of ordered differential fields. J. Symb. Log. 43(1), 82–91 (1978)

    Article  MathSciNet  Google Scholar 

  47. Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley and Los Angeles (1951)

    MATH  Google Scholar 

  48. van den Dries, L., Macintyre, A., Marker, D.: The elementary theory of restricted analytic fields with exponentiation. Ann. Math. (2) 140(1), 183–205 (1994)

    Article  MathSciNet  Google Scholar 

  49. van den Dries, L., Macintyre, A., Marker, D.: Logarithmic-exponential power series. J. Lond. Math. Soc. (2) 56(3), 417–434 (1997)

    Article  MathSciNet  Google Scholar 

  50. van den Dries, L., Macintyre, A., Marker, D.: In: Logarithmic-Exponential Series, Proceedings of the International Conference “Analyse & Logique”, Mons, 1997, vol. 111, pp. 61–113 (2001)

    MATH  Google Scholar 

  51. Wilkie, A.J.: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Am. Math. Soc. 9(4), 1051–1094 (1996)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Salma Kuhlmann.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kuhlmann, S. Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven: “Asymptotic Differential Algebra and Model Theory of Transseries”. Jahresber. Dtsch. Math. Ver. 120, 297–302 (2018). https://doi.org/10.1365/s13291-018-0179-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1365/s13291-018-0179-8

Navigation