Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T06:55:52.954Z Has data issue: false hasContentIssue false
  • English
  • Français

The theory of Liouville functions

Published online by Cambridge University Press:  12 March 2014

Pascal Koiran
Pascal Koiran*
Affiliation:
LIP, Ecole Normale Supérieure de Lyon, 46, Allée D'talie, 69364 Lyon Cedex 07, France, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A Liouville function is an analytic function H: ℂ → ℂ with a Taylor series such the an's form a “very fast growing” sequence of integers. In this paper we exhibit the complete first-order theory of the complex field expanded with H.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 2 , June 2003 , pp. 353 - 365
Copyright
Copyright © Association for Symbolic Logic 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Basu, S., Pollack, R., and Roy, M.-F., On the combinatorial and algebraic complexity of quantifier elimination. Journal of the Association for Computing Machinery, vol. 43 (1996), no. 6, pp. 1002–1045.CrossRefGoogle Scholar
[2]Blum, L., Cucker, F., Shub, M., and Smale, S., Complexity and real computation, Springer-Verlag, 1998.CrossRefGoogle Scholar
[3]Chapuis, O., Hrushovski, E., Koiran, P., and Poizat, B., La limite des thÉories de courbes génériques, this Journal, vol. 67 (2002), no. 1, pp. 24–34.Google Scholar
[4]Koiran, P., The limit theory of generic polynomials, Technical Report 01-35, LIP-Ecole Normale Supérieure Lyon, 2001, To appear in Proceedings of Logic Colloquium 2001.Google Scholar
[5]Koiran, P. and Portier, N., Back-and-forth systems for generic curves and a decision algorithm for the limit theory, Annals of Pure and Applied Logic, vol. 111 (2001), pp. 257–275.CrossRefGoogle Scholar
[6]Poizat, B., Le carré de l'égalité, this Journal, vol. 64 (1999), pp. 1339–1355.Google Scholar
[7]Poizat, B., Amalgames de Hrushovski: une tentative de classification, preprint, 2000.Google Scholar
[8]Renegar, J., On the computational complexity and geometry of the first-order theory of the reals, parts I, II, III, Journal of Symbolic Computation, vol. 13 (1992), no. 3, pp. 255–352.CrossRefGoogle Scholar
[9]Wilkie, A. J., Liouville functions, To appear in Proceedings of Logic Colloquium 2000.Google Scholar
[10]Zilber, B., Analytic and pseudo-analytic structures, To appear in Proceedings of Logic Colloquium 2000.Google Scholar
[11]Zilber, B., A theory of generic functions with derivatives, preprint.Google Scholar