Tabulation of cubic function fields via polynomial binary cubic forms

Rozenhart, Pieter; Jr., Michael Jacobson; Scheidler, Renate

doi:10.1090/S0025-5718-2012-02591-9

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.

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