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Tabulation of Cubic Function Fields Via Polynomial Binary Cubic Forms
ROZENHART, Pieter
Lithe and fast algorithmic number theory [LFANT]
Department of Mathematics and Statistics [Calgary]
Lithe and fast algorithmic number theory [LFANT]
Department of Mathematics and Statistics [Calgary]
ROZENHART, Pieter
Lithe and fast algorithmic number theory [LFANT]
Department of Mathematics and Statistics [Calgary]
< Reduce
Lithe and fast algorithmic number theory [LFANT]
Department of Mathematics and Statistics [Calgary]
Language
en
Article de revue
This item was published in
Mathematics of Computation. 2012, vol. 81, n° 280, p. 2335-2359
American Mathematical Society
English 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 ...Read more >
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.Read less <
Origin
Hal imported