On the computation of Hermite-Humbert constants for real quadratic number fields
type: Item
instance of: scholarly article
title: On the computation of Hermite-Humbert constants for real quadratic number fields
zbMATH Open document ID: 1098.11063
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2006-03-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/52738
review text: Let \(K/{\mathbb{Q}}\) be a totally real quadratic number field, with ring of integers \({\mathcal O}_K\) and discriminant \(d_K\). Let \(S=(S_1, S_2)\) be a positive real binary Humbert form over \(K\), where \(S_1\), \(S_2\) are positive definite \(2\times 2\) real matrices. In this paper, when the class number \(h_K = 1\), the authors use the theoretical background related to Humbert forms to develop an algorithm for computing extreme binary Humbert forms, i.e. those for which the Hermite-Humbert constant \(\gamma_K\) achieves a local maximum. Extreme forms have two properties: perfection and eutacticity. In 2001, \textit{R. Coulangeon} [see Euclidean lattices, spherical designs and modular forms. On the works of Boris Venkov. Genéve: L'Enseignement Mathématique. Monogr. Enseign. Math. 37, 147--162 (2001; Zbl 1139.11321)] proved that the number of extreme forms is finite up to unimodular equivalence. The main algorithm is divided into several subalgorithms that help to compute extreme Humbert forms, minimal vectors, all 3-sets, test equivalence of extreme Humbert forms,\dots The authors implement these algorithms in KANT/KASH and use them to determine the extreme Humbert forms for \({\mathbb{Q}}(\sqrt{13})\) and \({\mathbb{Q}}(\sqrt{17})\). In fact, for \({\mathbb{Q}}(\sqrt{13})\), they obtain 4000 triples, 770 Humbert forms with more than 4 minimal vectors, 3 Humbert forms with more than 4 minimal vectors up to unimodular equivalence. They compute the Hermite-Humbert constant \[ \gamma_{K, 2} = \sqrt{\frac{1476+144\sqrt{91}}{175}}=4.0353243\dots \] In the case of \({\mathbb{Q}}(\sqrt{17})\), the authors are not able to show that each extreme Humbert form has a unimodular pair of minimal vectors. But they compute extreme forms with pairs of unimodular minimal vectors. So they obtain 80000 triples, solve polynomial equations to determine Humbert forms, compute minimal vectors. They find two extreme Humbert forms giving \[ \gamma_{K}(S) = \sqrt{\frac{784+128\sqrt{3…
reviewed by: Alain Togbé
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11H55
zbMATH DE Number: 5016593
zbMATH Keywords: Algebraic number theory computations
zbMATH Keywords: quadratic extensions
author: Michael E. Pohst
author: Marcus Wagner
describes a project that uses: KANT/KASH
MaRDI profile type: Publication
OpenAlex ID: W2001015783
cites work: Hermite's Constant for Quadratic Number Fields
cites work: Q4654214
cites work: A Numerical Survey of the Floors of Various Hilbert Fundamental Domains
cites work: Q5509806
cites work: Théorie de la reduction des formes quadratiques definies positives dans un corps algébrique \(K\) fini
cites work: Hermite Constant and Extreme Forms for Algebraic Number Fields
DOI: 10.5802/JTNB.526
description: scientific article; zbMATH DE number 5016593