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Reduction in purely cubic function fields of unit rank one. (English) Zbl 1035.11057
Bosma, Wieb (ed.), Algorithmic number theory. 4th international symposium. ANTS-IV, Leiden, the Netherlands, July 2–7, 2000. Proceedings. Berlin: Springer (ISBN 3-540-67695-3). Lect. Notes Comput. Sci. 1838, 515-532 (2000).
The author gives a complete analysis of the algorithm used to obtain a reduced basis of a fractional ideal in a purely cubic function field of unit rank one. This algorithm has been presented and used by the author [J. Théor. Nombres Bordx. 13, 609–631 (2001; Zbl 0995.11064)] to study the ideal arithmetic and infrastructure in such function fields. The author answers questions concerning the complexity of a reduction step, the precision required in the truncation of a Puiseux series, and the size of the quantities involved in the algorithm.
For the entire collection see [Zbl 0960.00039].

11R58 Arithmetic theory of algebraic function fields
11Y16 Number-theoretic algorithms; complexity
11R16 Cubic and quartic extensions
14H05 Algebraic functions and function fields in algebraic geometry