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Tabulation of Cubic Function Fields Via Polynomial Binary Cubic Forms

hal.structure.identifierLithe and fast algorithmic number theory [LFANT]
hal.structure.identifierDepartment of Mathematics and Statistics
dc.contributor.authorROZENHART, Pieter
hal.structure.identifierDepartment of Computer Science [Calgary] [CPSC]
dc.contributor.authorJACOBSON JR., Michael
hal.structure.identifierDepartment of Mathematics and Statistics
dc.contributor.authorSCHEIDLER, Renate
dc.date.accessioned2024-04-04T02:29:06Z
dc.date.available2024-04-04T02:29:06Z
dc.date.created2010-04-27
dc.date.issued2012
dc.identifier.issn0025-5718
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/190147
dc.description.abstractEnWe 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.
dc.language.isoen
dc.publisherAmerican Mathematical Society
dc.title.enTabulation of Cubic Function Fields Via Polynomial Binary Cubic Forms
dc.typeArticle de revue
dc.identifier.doi10.1090/S0025-5718-2012-02591-9
dc.subject.halMathématiques [math]/Théorie des nombres [math.NT]
dc.identifier.arxiv1004.4785
bordeaux.journalMathematics of Computation
bordeaux.page2335-2359
bordeaux.volume81
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue280
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierinria-00477111
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//inria-00477111v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Mathematics%20of%20Computation&rft.date=2012&rft.volume=81&rft.issue=280&rft.spage=2335-2359&rft.epage=2335-2359&rft.eissn=0025-5718&rft.issn=0025-5718&rft.au=ROZENHART,%20Pieter&JACOBSON%20JR.,%20Michael&SCHEIDLER,%20Renate&rft.genre=article


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