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Journal Articles Mathematics of Computation Year : 2012

Tabulation of Cubic Function Fields Via Polynomial Binary Cubic Forms

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 $\mathbb_{q}^*$, up to a given bound $X$ on $|D| = q^{\deg(D)}$. 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 $O(q^4 X^{1+\epsilon})$ field operations when $D$ has odd degree, and $O(q^5 X^{1+\epsilon})$ when $D$ has even degree. It performs quite well in practice. The algorithm, examples and numerical data for $q=5,7,11,13$ are included.

Dates and versions

inria-00477111 , version 1 (28-04-2010)

Identifiers

Cite

Pieter Rozenhart, Michael Jacobson Jr., Renate Scheidler. Tabulation of Cubic Function Fields Via Polynomial Binary Cubic Forms. Mathematics of Computation, 2012, 81 (280), pp.2335-2359. ⟨10.1090/S0025-5718-2012-02591-9⟩. ⟨inria-00477111⟩
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