## Tabulation of cubic function fields via polynomial binary cubic forms

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- by Pieter Rozenhart, Michael Jacobson Jr. and Renate Scheidler;
- Math. Comp.
**81**(2012), 2335-2359 - DOI: https://doi.org/10.1090/S0025-5718-2012-02591-9
- 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.## References

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## Bibliographic 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
- MR Author ID: 308756
- ORCID: 0000-0001-7164-8769
- Email: rscheidl@math.ucalgary.ca
- Received by editor(s): April 27, 2010
- Received by editor(s) in revised form: May 30, 2011
- Published electronically: March 14, 2012
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**81**(2012), 2335-2359 - MSC (2010): Primary 11Y40, 11R16; Secondary 11R58
- DOI: https://doi.org/10.1090/S0025-5718-2012-02591-9
- MathSciNet review: 2945159