Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T14:12:44.348Z Has data issue: false hasContentIssue false
  • English
  • Français

Rational solutions of linear differential equations

Part of: Sequences and sets

Published online by Cambridge University Press:  09 April 2009

J.-P. Bezivin  and
P. Robba
J.-P. Bezivin
Affiliation:
Université ParisVI Math. Tour 45-46, 5ème étage 4, place Jussieu 75230 Paris Cedex 05, France
P. Robba
Affiliation:
Université Paris-Sud Math.Bâtiment 425 91405 Orsay Cedex, 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.

Let L be a linear differential operator with rational coefficients such that 0 is not an irregular singularity of L and that for sufficiently many p's the equation Lv = 0 has no zero solution mod p. We show that if u is a formal power series whose coefficients are p-adic integers for almost all p and if Lu is rational, then u too is rational.

MSC classification

Secondary: 11B37: Recurrences
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 2 , April 1989 , pp. 184 - 196
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Amice, Y., Les nombres p-adiques (PUF Collection Sup 1977).Google Scholar
[2]Bezivin, J.-P., ‘Une propriété arithmétique de certains opérateurs différentiels’, Manuscripta Math. 57 (1987), 351–372.CrossRefGoogle Scholar
[3]Cantor, D., ‘On arithmetic properties of coefficients of rational function, Pacific J. Math. 15 (1965), 55–58.CrossRefGoogle Scholar
[4]Clark, D., ‘A note on the p–adic convergence of linear differential equations’, Proc. Amer. Math. Soc. 17 (1966), 262–269.Google Scholar
[5]Dwork, B. and Robba, P., ‘On ordinary linear p–adic differential equations’, Trans. Amer. Math. Soc. 231 (1977), 1–46.Google Scholar
[6]Honda, T., ‘Algebraic differential equations’, Symposia Math. 24 (1981), 169–204.Google Scholar
[7]Janusz, G., Algebraic number fields (Academic Press, New York, 1973).Google Scholar
[8]Pólya, G., ‘Arithmetische Eigenschaften der Reiherenwichlungen rationaler Funktionen’, J. Reine Angew. Math. 151 (1921), 1–31.CrossRefGoogle Scholar
[9]Robba, P., ‘Prolongement des solutions d'une équations différentielle p–adique’, C. R. Acad. Sci. Paris 279 (1974), 153–154.Google Scholar
[10]Robba, P., ‘On the index of p–adic differential operators II’, Duke Math. J. 43 (1976), 19–31.CrossRefGoogle Scholar