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Transcendence measures by a method of Mahler

  • William Miller (a1)

Abstract

Suppose that f(z) is a function of one complex variable satisfying

where ρ is an integer larger than 1 and a(z) and b(z) are rational functions. We consider f evaluated at the algebraic point a and develop a transcendence measure for f(α) under suitable conditions on f and α.

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Copyright

References

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Lang, S. (1965), Algebra (Addison-Wesley, Reading, Massachusetts).
Lang, S. (1966), Introduction to transcendental numbers (Addison-Wesley, Reading, Massachusetts).
Loxton, J. H. and van der Poorten, A. J. (1976), ‘On algebraic functions satisfying a class of functional equations’, Aequations Math. 14, 413420.
Loxton, J. H. and van der Poorten, A. J. (1977), ‘Transcendence and algebraic independence by a method of Mahler’, Transcendence Theory-Advances and Applications, edited by Baker, A. and Masser, D. W., Chapter 15, pp. 211226 (Academic Press).
Kubota, K. K. (1977), ‘On the algebraic independence of holomorphic solutions of certain functional equations and their values’, Math. Ann. 227, 950.
Mahler, K. (1929), ‘Arithmetische Eigenschaften der Losungen einer Klasse von Funktionalgleichungen’, Math. Ann. 101 342366.
Miller, W. (1979), Transcendence measures for values of analytic solutions to certain functional equations (Ph.D. Thesis, University of Michigan).
Waldschmidt, M. (1974), Nombres transcendants, Lecture Notes in Mathematics, 402 (Springer, Berlin).
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