Arithmetic Gevrey series and transcendence. A survey

Yves André

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 1, page 1-10
  • ISSN: 1246-7405

Abstract

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We review the main results of the theory of arithmetic Gevrey series introduced in [3] [4], their applications to transcendence, and a few diophantine conjectures on the summation of divergent series.

How to cite

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André, Yves. "Arithmetic Gevrey series and transcendence. A survey." Journal de théorie des nombres de Bordeaux 15.1 (2003): 1-10. <http://eudml.org/doc/249075>.

@article{André2003,
abstract = {We review the main results of the theory of arithmetic Gevrey series introduced in [3] [4], their applications to transcendence, and a few diophantine conjectures on the summation of divergent series.},
author = {André, Yves},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {1},
pages = {1-10},
publisher = {Université Bordeaux I},
title = {Arithmetic Gevrey series and transcendence. A survey},
url = {http://eudml.org/doc/249075},
volume = {15},
year = {2003},
}

TY - JOUR
AU - André, Yves
TI - Arithmetic Gevrey series and transcendence. A survey
JO - Journal de théorie des nombres de Bordeaux
PY - 2003
PB - Université Bordeaux I
VL - 15
IS - 1
SP - 1
EP - 10
AB - We review the main results of the theory of arithmetic Gevrey series introduced in [3] [4], their applications to transcendence, and a few diophantine conjectures on the summation of divergent series.
LA - eng
UR - http://eudml.org/doc/249075
ER -

References

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  1. [1] Y. André, Théorie des motifs et interprétation géométrique des valeurs p-adiques de G-fonctions (une introduction). Number theory (Paris, 1992-1993), 37-60, London Math. Soc. Lecture Note Ser., 215, Cambridge Univ. Press, Cambridge, 1995. Zbl0848.11034MR1345171
  2. [2] Y. André, G-fonctions et transcendance. J. Reine Angew. Math.476 (1996), 95-125. Zbl0848.11033MR1401697
  3. [3] Y. André, Séries Gevrey de type arithmétique I. Théorèmes de pureté et de dualité. Ann. of Math.151 (2000), 705-740. Zbl1037.11049MR1765707
  4. [4] Y. André, Séries Gevrey de type arithmétique II. Transcendance sans transcendance. Ann. of Math.151 (2000), 741-756. Zbl1037.11050MR1765708
  5. [5] D. Bertrand, On André's proof of the Siegel-Shidlovsky theorem. Colloque Franco-Japonais: Théorie des Nombres Transcendants (Tokyo, 1998), 51-63, Sem. Math. Sci., 27, Keio Univ., Yokohama, 1999. MR1726524
  6. [6] F. Beukers, Algebraic values of G-fonctions. J. Reine Angew. Math.434 (1993), 45-65. Zbl0753.11024MR1195690
  7. [7] L. Euler, De seriebus divergentibus. In Opera omnia I. 14 Teubner (1925), 601-602. 
  8. [8] M. Gevrey, La nature analytique des solutions des équations aux dérivées partielles. Ann. Sci. École Norm. Sup. (3) 25 (1918), 129-190. Zbl46.0721.01MR1509208JFM46.0721.01
  9. [9] E. Maillet, Sur les séries divergentes et les équations différentielles. Ann. Sci. École Norm. Sup. (1903), 487-518. Zbl34.0282.01MR1509033JFM34.0282.01
  10. [10] O. Perron, Über lineare Differentialgleichungen mit rationalen Koeffizienten. Acta math.34 (1910), 139-163. Zbl42.0329.02JFM42.0329.02
  11. [11] J.-P. Ramis, Séries divergentes et Théories asymptotiques. Panoramas et Synthèses21, SMFParis, 1993. Zbl0830.34045MR1272100
  12. [12] A. Shidlovsky, Transcendental Numbers. de Gruyter Studies in Math. 12, 1989; translation of Transtsendentnye Chisla, Nauka, 1987. Zbl0689.10043MR911106
  13. [13] C.L. Siegel, Über einige Anwendungen diophantischer Approximationen. Abh. Preuss. Akad. Wiss. Phys.-Math. Kl.1 (1929), 1-70. Zbl56.0180.05JFM56.0180.05
  14. [14] G. Watson, A theory of Asymptotic series. Philos. Trans. Roy. Soc. London Ser. A211 (1911), 279-313. Zbl42.0273.01JFM42.0273.01

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