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Effectivity in independence measures for values of E-functions

Published online by Cambridge University Press:  09 April 2009

W. Dale Brownawell
W. Dale Brownawell
Affiliation:
Department of MathematicsPennsylvania State University, University Park, Pennsylvania 16802, U.S.A.
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Abstract

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We establish a measure of algebraic independence for values of E-functions which is more nearly effectively computable than the previous one. When the system of equations meets either of two criteria, then the measure becomes entirely effectively computable.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 39 , Issue 2 , October 1985 , pp. 227 - 240
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Bertrand, D. and Beukers, F., ‘Equations différentielles linéaires et majorations de multiplicités’, to appear.Google Scholar
[2]Brownawell, W. D., ‘Travaux récents de Ju. V. Nesterenko’, Séminaire Delange-Pisot-Poitou, 1977/1978, no. 35, 6 p.Google Scholar
[3]Feldman, N. I. and Shidlovsky, A. B., ‘The development and present state of the theory of transcendental numbers’, Uspehi Mat. Nauk 22 No. 3 (1967), 3–81. English transl.Google Scholar
Russian Math. Surveys 22 No. 3 (1967), 1–79.Google Scholar
[4]Lang, S., ‘A transcendence measure for E-functions’, Mathematika 9 (1962), 157–161.CrossRefGoogle Scholar
[5]Mahler, K., ‘Zur Approximation der Exponentialfunktion und des Logarithmus. I’ J. Reine Angew. Math. 166 (1932), 118–136.CrossRefGoogle Scholar
[6]Mahler, K., Lectures on transcendental numbers (Lecture Notes in Mathematics, No. 546, Springer-Verlag, Berlin, Heidelberg, New York, 1976).CrossRefGoogle Scholar
[7]Nesterenko, Yu. V., ‘On the algebraic dependence of components of solutions of systems of linear differential equations’, Izv. Akad. Nauk SSSR, Ser. Mat. 38 (1974), 495–512. English transl.Google Scholar
Math. USSR Izv. 8 (1974), 501–518.CrossRefGoogle Scholar
[8]Nesterenko, Yu. V., ‘Bounds on the order of zeros of a class of functions and their application to the theory of transcendental numbers’, Izv. Akad. Nauk SSSR, Ser. Mat 41 (1977), 239–270. English transl.Google Scholar
Math. USSR Izv. 11 (1977), 253–284.Google Scholar
[9]Shidlovsky, A. B., ‘A criterion for algebraic independence of the values of a class of entire functions’, Izv. Akad. Nauk SSSR Ser. Mat. 23 (1959), 35–66. English transl.Google Scholar
Amer. Math. Soc. Transl. (2) 22 (1962), 339–370.Google Scholar
[10]Shidlovsky, A. B., ‘On arithmetic properties of values of analytic functions’, Trudy MIAN SSSR Steklova 132 (1973), 169–202. English transl.Google Scholar
Proceedings of the Steklov Institute of Math., No. 132 (1973), 193–233.Google Scholar
[11]Siegel, C. L., ‘Über einige Anwendungen diophantischer Approximationen’, Abh. Preuss. Akad. Wiss. Phys.-mat. Kl. Berlin (1929), reprinted inGoogle Scholar
Carl Ludwig Siegel Gesammelte Abhandlungen, I. Springer-Verlag, Berlin, Heidelberg, New York, 1966, 209–266.Google Scholar
[12]Siegel, C. L., ‘Transcendental numbers’, Ann. of Math. Studies 16 (Princeton University Press, Princeton, N. J., 1949).Google Scholar
[13]Tai, N. T., ‘On estimates for the orders of zeros of polynomials in analytic functions and their application to estimates for the relative transcendence measure of values of E-functions’, Mat. Sb. 120 (1983), 112–142. English transl.Google Scholar
Math. USSR Sb. 48 (1984), 111–140.CrossRefGoogle Scholar