ROZENHART, Pieter; JACOBSON JR., Michael; SCHEIDLER, Renate

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

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ROZENHART, Pieter
Lithe and fast algorithmic number theory [LFANT]
Department of Mathematics and Statistics

JACOBSON JR., Michael
Department of Computer Science [Calgary] [CPSC]

SCHEIDLER, Renate
Department of Mathematics and Statistics

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Mathematics of Computation. 2012, vol. 81, n° 280, p. 2335-2359

American Mathematical Society

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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.< Réduire

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Tabulation of Cubic Function Fields Via Polynomial Binary Cubic Forms

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