Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-25T13:00:29.178Z Has data issue: false hasContentIssue false
  • English
  • Français

Fractional parts of Linear polynomials and an application to hypergeometric functions

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  09 April 2009

Roberto Dvornicich  and
Umberto Zannier
Roberto Dvornicich
Affiliation:
Dipartimento di Matematica via Buonarroti, 2 56127 PisaItaly e-mail: dvornic@dm.unipi.it
Umberto Zannier
Affiliation:
Istituto Universitario di Architettura D.C.A.S.Croce, 191 (Tolentini) 30135 VeneziaItaly e-mail: zannier@brezza.iuav.unive.it
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.

Using a result on arithmetic progressions, we describe a method for finding the rational h–tuples ρ = (ρl,…,ρh) such that all the multiples mρ (for m coprime to a denominator of ρ) lie in a linear variety modulo Z. We give an application to hypergeometric functions.

MSC classification

Secondary: 11J54: Small fractional parts of polynomials and generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 70 , Issue 3 , June 2001 , pp. 401 - 424
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Baldassarri, F. and Dwork, B., ‘On second order linear differential equations with algebraic solutions’, Amer. J. Math. 101 (1979), 42–76.CrossRefGoogle Scholar
[2]Beukers, F. and Heckman, G., ‘Monodromy for the hypergeometric function nFn–1’, Invent. Math. 95 (1989), 325–354.CrossRefGoogle Scholar
[3]Davenport, H. and Schinzel, A., ‘Diophantine approximation and sums of roots of unity’, Math. Ann. 169 (1967), 118–135.CrossRefGoogle Scholar
[4]Dvornicich, R. and Zannier, U., ‘On sums of roots of unity’, Monatsh. Math. 129 (2000), 97–108.CrossRefGoogle Scholar
[5]Errera, A., ‘Zahlentheoretische Lösung einer functionentheoretischen Frage’, Rend. Circ. Mat. Palermo 35 (1913), 107–144.CrossRefGoogle Scholar
[6]Hardy, G. H. and Wright, E. M., Introduction to the theory of numbers (Oxford at Clarendon Press, 1979).Google Scholar
[7]Jones, A. J., ‘Cyclic overlattices (I)’, Acta Arith. XVII (1970), 303–314.CrossRefGoogle Scholar
[8]Jones, A. J., ‘Cyclic overlattices (II)’, Acta Arith. XVIII (1971), 93–103.CrossRefGoogle Scholar
[9]Katz, N. M., ‘Algebraic solutions of differential equations (p–curvature and the Hodge filtration)’, Invent. Math. 18 (1972), 1–118.CrossRefGoogle Scholar
[10]Katz, N. M., ‘A conjecture in the arithmetic theory of differential equations’, Bull. Soc. Math. France 110 (1982), 203–239.CrossRefGoogle Scholar
[11]Landau, E., ‘Eine Anwendung des Eisensteinschen Satzes auf die Theorie der Gaussschen Differentialgleichung’, J. Reine Angew. Math. 127 (1904), 92–102.Google Scholar
[12]Landau, E., ‘Über einen zahlentheoretischen Satz und seine Anwendung auf die hypergeometrische Reihe’, S.-B. Heidelberger Akad. Wiss. 18 (1911), 3–38.Google Scholar
[13]Matsuda, M., Lectures on algebraic solutions of hypergeometric differential equations (Dept. of Mathematics, Kyoto Univ., Kinokuniya Co., Ltd., 1985).Google Scholar
[14]Milnor, J., ‘On polylogarithms, Hurwitz zeta-functions, and the Kubert identities’, Enseign. Math. 29 (1983), 281–322.Google Scholar
[15]Schwarz, H. A., ‘Über diejenigen Fälle, in welchen die gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt’, J. Reine Angew. Math. 75 (1873), 292–335.Google Scholar