Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-25T10:53:37.187Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note On the Hadamard Kth Root of a Rational Function

Published online by Cambridge University Press:  09 April 2009

Roberts Rumely  and
A. J. Van der Poorten
Roberts Rumely
Affiliation:
Department of MathematicsUniversity of GeorgiaAthens, Georgia 30602, U.S.A.
A. J. Van der Poorten
Affiliation:
School of Mathematics and PhysicsMacquarie UniversityNorth Ryde, N.S.W. 2113, Australia
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.

Suppose the sequence of Taylor coefficients of a rational function f consists of kth powers of elements all belonging to some finitely generated extension field F of Q. Then it is a generalisation of a conjecture of Pisot that there is a rational function with Taylor coefficients term-by-term kth roots of those of f. The authors show that it suffices to prove the conjecture in the case that the field of definition is a number field and prove the conjecture in that case subject to the constraint that f has a dominant pole, that is, that there is a valuation with respect to which f has a unique pole either of maximal or of minimal absolute value.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 3 , December 1987 , pp. 314 - 327
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Artin, E. and Tate, J., Class field theory (Harvard University).Google Scholar
[2]Benzaghou, Benali, ‘Algèbres de Hadamard’, Bull. Soc. Math. France 98 (1970), 209–252.CrossRefGoogle Scholar
[3]Bézivin, J.-P., ‘Factorisation de suites récurrentes linéaires’, Groupe d'élude d'analyse ultramétrique (Amice-Barsky-Robba) 1979–1981 expose 33, pp. 6–9 (Institut Henri Poincaré, Paris).Google Scholar
[4]Bézivin, J.-P., ‘Factorisation de suites récurrentes linéaires et applications’, Bull. Soc. Math. France 112 (1984), 365–376.CrossRefGoogle Scholar
[5]Bombieri, E., ‘On G-functions’, Recent progress in analytic number theory, Vol. 2, edited by Halberstam, H. and Hooley, C., pp. 1–67 (Academic Press, London, 1981).Google Scholar
[6]Cantor, David G., ‘On arithmetic properties of the Taylor series of rational functions II’, Pacific J. Math. 41 (1972), 329–334.CrossRefGoogle Scholar
[7]Perelli, A. and Zannier, U., ‘Arithmetic properties of certain recurrence sequences’, J. Austral. Math. Soc.(Ser. A) 37 (1984), 4–16.CrossRefGoogle Scholar
[8]Pisot, Charles, Conférences donnees à I'Institut Fourier de Grenoble,(1959).Google Scholar
[9]Rumely, Robert S. and van der Poorten, A. J., ‘Remarks on generalised power sums’, Bull. Austral. Math. Soc.Google Scholar
[10]Salem, Raphael, Algebraic numbers and Fourier analysis, (Heath, Boston, Mass., 1962).Google Scholar
[11]van der Poorten, A. J., ‘Some problems of recurrent interest’, Colloq. Math. Soc. Janos Bolyai 34 Topics in classical number theory, 1265–1294.Google Scholar