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On meromorphic functions of one complex variable having algebraic Laurent coefficients

Published online by Cambridge University Press:  17 April 2009

Daniel Bertrand  and
Michel Waldschmidt
Daniel Bertrand
Affiliation:
Université de Nice, Mathématiques - Parc Valrose, 06034 Nice Cedex, France;
Michel Waldschmidt
Affiliation:
Université Pierre et Marie Curie, Mathématiques T. 45–46, 75230 Paris Cedex 05, France.
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Abstract

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We study the set of points at which two algebraically independent meromorphic functions have algebraic coefficients in their Laurent expansions. After a survey of the present knowledge in this field, we obtain two general transcendence criteria which sharpen previous results of Straus, Schneider and Lang. As a corollary, we give a new proof, based on Gel'fond's method, of some of Siegel's results on E-functions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 24 , Issue 2 , October 1981 , pp. 247 - 267
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Bertrand, Daniel, “Équations différentielles algébriques et nombres transcendants dans les domaines complexes et p-adiques” (Thèse, Université de Paris VI, Paris, 1975).Google Scholar
[2]Bertrand, Daniel, “A transcendence criterion for meromorphic functions”, Transcendence theory: advances and applications, 187–193 (Proc. Conf. Cambridge,1976.Academic Press,London, New York, San Francisco, 1977).Google Scholar
[3]Bertrand, Daniel, “Un théorème de Schneider-Lang sur certains domaines non simplement connexes”, Séminaire Delange-Pisot-Poitou (16e année: 1974/75), Théorie des nombres, Fasc. 2, Exp. No. G18 (Secrétariet Mathématique, Paris, 1975).Google Scholar
[4]Choodnovsky, G.V., “A new method for the investigation of arithmetical properties of analytic functions”, Ann. of Math. (2) 109 (1979), 353–376.CrossRefGoogle Scholar
[5]Choodnovsky, Gregory V., “Values of meromorphic functions of order 2”, Séminaire Delange-Pisot-Poitou (19e année: 1977/78), Théorie des nombres, Fasc. 2, Exp. No. 45 (Secrétariet Mathématique, Paris, 1978).Google Scholar
[6]Chudnovsky, G.V., “Singular points on complex hypersurfaces and multidimensional Schwarz lemma”, Séminaire de théorie des nombres, Paris 1979−80, 29−70 (Séminaire Delange-Pisot-Poitou. Progress in Mathematics, 12. Birkhäuser, Boston, Basel, 1981).Google Scholar
[7]Gramain, François, “Fonctions entières arithmétiques”, Séminaire Pierre Lelong - Henri Skoda (Analyse), 17e année, 1976/77, 96–125 (Lecture Notes in Mathematics, 694. Springer–Verlag, Berlin, Heidelberg, New York, 1978).CrossRefGoogle Scholar
[8]Gramain, François, “Fonctions entières arithmétiques”, Séminaire Delange-Pisot-Poitou (19e année: 1977/78), Théorie des nombres, Fasc. 1, Exp. No. 8 (Secrétariet Mathématique, Paris, 1978).Google Scholar
[9]Gross, Fred, “Entire functions of several variables with algebraic derivatives at certain algebraic points”. Pacific J. Math. 31 (1969), 693–701.CrossRefGoogle Scholar
[10]Lang, Serge, “Transcendental points on group varieties”, Topology 1 (1962), 313–318.CrossRefGoogle Scholar
[11]Lang, Serge, Introduction to transcendental numbers (Addison-Wesley, Reading, Massachusetts; London; Don Mills, Ontario; 1966).Google Scholar
[12]Mahler, Kurt, Lectures on transcendental numbers (Lecture Notes in Mathematics, 546. Springer-Verlag, Berlin, Heidelberg, New York, 1976).CrossRefGoogle Scholar
[13]Pólya, G., “Über die kleinsten ganzen Funktionen, deren sämtliche Derivierte im Punkte k = 0 ganzzahlig sind”, Tôhoku Math. J. 19 (1921), 65–68.Google Scholar
[14]Reyssat, Éric, “Travaux récents de G. V. Čudnovskij”, Séminaire Delange-Pisot-Poitou (18e année: 1976/77), Théorie des nombres, Fasc. 2, Exp. No. 29 (Secrétariet Mathématique, Paris, 1977).Google Scholar
[15]Schneider, Theodor, “Ein Satz über ganzwertige Funktionen als Prinzip für Transzandenzbeweise”, Math. Ann. 121 (1949), 131–140.CrossRefGoogle Scholar
[16]Schneider, Theodor, Einführung in die transzendenten Zahlen (Die Grundlehren der mathematischen Wissenschaften, 81. Springer–Verlag, Berlin, Göttingen, Heidelberg, 1957).CrossRefGoogle Scholar
[17]Siegel, C.L., “Ueber einige Anwendungen diophantisher Approximationen”, Abh. Preuss. Akad. Wiss. Phys.-Math. Kl. (1929), No. 1. See also: “Über einige Anwendungen diophantisher Approximation”, Gesammelte Abhandlungen, Band I, 209–241 (Springer-Verlag, Berlin, Heidelberg, New York, 1966).Google Scholar
[18]Siegel, Carl Ludwig, Transcendental numbers (Annals of Mathematics Studies, 16. Princeton University Press, Princeton, 1949).Google Scholar
[19]Straus, E.G., “On entire functions with algebraic derivatives at certain algebraic points”, Ann. of Math. (2) 52 (1959), 188–198.CrossRefGoogle Scholar
[20]Waldschmidt, Michel, “On functions of several variables having algebraic Taylor coefficients”, Transcendence theory: advances and applications, 169–186 (Proc. Conf. Cambridge,1976.Academic Press,London, New York, San Francisco, 1977).Google Scholar
[21]Waldschmidt, Michel, Nombres transcendants et groupes algébriques (Astérisque, 69–70. Société Mathématique de France, Paris, 1979).Google Scholar