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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Tabulation of cubic function fields via polynomial binary cubic forms


Authors: Pieter Rozenhart, Michael Jacobson Jr. and Renate Scheidler
Journal: Math. Comp. 81 (2012), 2335-2359
MSC (2010): Primary 11Y40, 11R16; Secondary 11R58
Published electronically: March 14, 2012
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Abstract: We present a method for tabulating all cubic function fields over $ \mathbb{F}_q(t)$ whose discriminant $ D$ has either odd degree or even degree and the leading coefficient of $ -3D$ is a non-square in $ \mathbb{F}_{q}^*$, up to a given bound $ B$ on $ \deg (D)$. Our method is based on a generalization of Belabas' method for tabulating cubic number fields. The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms. The algorithm requires $ O(B^4 q^B)$ field operations as $ B \rightarrow \infty $. The algorithm, examples and numerical data for $ q=5,7,11,13$ are included.


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Additional Information

Pieter Rozenhart
Affiliation: Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada
Email: pmrozenh@alumni.uwaterloo.ca

Michael Jacobson Jr.
Affiliation: Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada
Email: jacobs@cpsc.ucalgary.ca

Renate Scheidler
Affiliation: Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada
Email: rscheidl@math.ucalgary.ca

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02591-9
PII: S 0025-5718(2012)02591-9
Keywords: Computational number theory, cubic function fields, field tabulation
Received by editor(s): April 27, 2010
Received by editor(s) in revised form: May 30, 2011
Published electronically: March 14, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.