Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-25T05:00:38.406Z Has data issue: false hasContentIssue false

On meromorphic functions of one complex variable having algebraic Laurent coefficients

Published online by Cambridge University Press:  17 April 2009

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
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, 187193 (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), 353376.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, 96125 (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), 693701.CrossRefGoogle Scholar
[10]Lang, Serge, “Transcendental points on group varieties”, Topology 1 (1962), 313318.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), 6568.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), 131140.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, 209241 (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), 188198.CrossRefGoogle Scholar
[20]Waldschmidt, Michel, “On functions of several variables having algebraic Taylor coefficients”, Transcendence theory: advances and applications, 169186 (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