Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T02:35:32.790Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-periodic continued fractions in hyperelliptic function fields

Published online by Cambridge University Press:  17 April 2009

Alfred J. van der Poorten
Alfred J. van der Poorten
Affiliation:
ceNTRe for Number Theory Research, Macquarie University, Sydney, NSW 2109, Australia, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

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.

Dedicated to George Szekeres on his 90th birthday

We discuss the exponential growth in the height of the coefficients of the partial quotients of the continued fraction expansion of the square root of a generic polynomial.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 2 , October 2001 , pp. 331 - 343
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Adams, W.W. and Razar, M.J., ‘Multiples of points on elliptic curves and continued fractions’, Proc. London Math. Soc. 41 (1980), 481–498.CrossRefGoogle Scholar
[2]Berry, T.G., ‘On periodicity of continued fractions in hyperelliptic function fields’, Arch. Math. (Basel) 55 (1990), 259–266.CrossRefGoogle Scholar
[3]Bombieri, E. and Cohen, P.B., ‘Siegel's lemma, Padé approximations and Jacobians’, with an appendix by Umberto Zannier and dedicated to Enzio De Giorgi, Ann. Scuola Norm, Sup. Pisa Cl. Sci. (4) 25 (1997), no 1–2, (1998), 155–178.Google Scholar
[4]Cantor, D.G., ‘On the continued fractions of quadratic surds’, Acta Arith. 68 (1994), 295–305.CrossRefGoogle Scholar
[5]Davenport, J.H., On the integration of algebraic functions, Lecture Notes in Computer Science 102 (Springer-Verlag, Berlin, Heidelberg, New York, 1981).CrossRefGoogle Scholar
[6]Elkies, N., ‘Simple genus-2 Jacobians with high-order torsion points’, (e-mail, 01, 2001).Google Scholar
[7]Friesen, C., ‘Continued fraction characterization and generic ideals’, in The arithemtic of function fields, Ohio State University Mathematical Research Institute Publications 2, (Proceedings of the workshop held at The Ohia State University,Ohio,June 17–26, 1991) (Walter de Gruyter & Co., Berlin, 1992), pp. 465–474.Google Scholar
[8]Goss, D., Hayes, D.R. and Rosen, M.I., Editors, The arithmetic of function fields, Ohio State University Mathematical Research Institute Publications 2, (Proceedings of the workshop held at The Ohio State University,Columbus, Ohio,June 17–26, 1991) (Walter de Gruyter & Co., Berlin, 1992).CrossRefGoogle Scholar
[9]Knuth, D.E., The art of computer programming 2, Seminumerical Algorithms, (2nd printing), 1969.Google Scholar
[10]van der Poorten, A., ‘Formal power series and their continued fraction expansion’, in Algorithmic Number Theory, (Buhler, Joe, Editor), Lecture Notes in Computer Science 1423, (Proc. Third International Symposium,ANTS-III,Portland, Oregon,June 1998) (Springer-Verlag, Berlin, Heidelberg, New York, 1998), pp. 358–371.CrossRefGoogle Scholar
[11]van der Poorten, A.J. and Tran, X.C., ‘Quasi-elliptic integrals and periodic continued fractions’, Monatsh. Math. 131 (2000), 155–169.CrossRefGoogle Scholar
[12]Schmidt, W.M., ‘On continued fractions and diophantine approximation in power series field’, Acta Arith. 95 (2000), 139–166.CrossRefGoogle Scholar
[13]Tran, X.C., Periodic continued fractions in function fields, Ph.D. Thesis (Macquarie University, Sydney, 2000).Google Scholar
[14]Yu, J., ‘Arithmetic of hyperelliptic curves’, in Aspects of Mathematics (Hang kong University, 1999).Google Scholar