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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: Pascal.Koiran@ens-lyon.fr
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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

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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