An Explicit Construction of Initial Perfect Quadratic Forms over Some Families of Totally Real Number Fields

Leibak, Alar

doi:10.1007/978-3-540-79456-1_16

Cited by 3 publications

(3 citation statements)

References 10 publications

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“…For F real quadratic and n = 2, we carry out our construction explicitly to compute all inequivalent binary perfect forms for F = Q( √ d), d ≤ 66. These results complement work of Leibak and Ong [Lei08,Lei05,Ong86].…”

Section: Introductionsupporting

confidence: 89%

Perfect forms over totally real number fields

Gunnells,

Yasaki

2009

Preprint

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A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f (v) = m. This concept was introduced by Voronoï and later generalized by Koecher to arbitrary number fields. One knows that up to a natural "change of variables" equivalence, there are only finitely many perfect forms, and given an initial perfect form one knows how to explicitly compute all perfect forms up to equivalence. In this paper we investigate perfect forms over totally real number fields. Our main result explains how to find an initial perfect form for any such field. We also compute the inequivalent binary perfect forms over real quadratic fields Q( √ d) with d ≤ 66.

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Section: Introductionsupporting

confidence: 89%

Perfect forms over totally real number fields

Gunnells,

Yasaki

2009

Preprint

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“…. , n (see [7]). Throughout the paper, we will simply refer to positive definite quadratic forms as quadratic forms, unless we need to clarify.…”

Section: Introductionmentioning

confidence: 99%

Unit Reducible Fields and Perfect Unary Forms

Leibak,

Porter,

Ling

2024

Journal De Théorie Des Nombres De Bordeaux

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“…There exists an algorithm to compute all of the GL n (O)-equivalence classes of perfect n-ary quadratic forms over F once an initial perfect form is found [15,16]. This is investigated in the totally real number field case in [13,17]. We remark that a different notion of perfection has been investigated in [3,6,20].…”

Section: Introductionmentioning

confidence: 99%