Lamination and antilamination of Euclidean lattices
type: Item
instance of: scholarly article
title: Lamination and antilamination of Euclidean lattices
zbMATH Open document ID: 1204.11102
author: Marc Gindraux
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2010-08-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10898
review text: The author studies some invariants related to the Hermite-Korkine-Zolotareff reduction of Euclidean lattices (or of positive definite quadratic forms). A real positive definite quadratic form is said \textit{laminated} if its decomposition as a sum of squares \[ {}^t xAx= A_1(x_1+\alpha_{12} x_2+\dots\alpha_{1n}x_n)^2+A_2(x_2+\alpha_{23}x_3+\dots+\alpha_{2n}x_n)^2+\dots+A_nx_n^2, \] is obtained by successive minimizations of coefficients \(A_1, A_2,\dots, A_{n-1}\). The invariant \[ \gamma_-''(A)=\min_{\text{HKZ}(\Lambda)}{\sqrt{A_1/A_n}}, \] (as the historic Korkine-Zolotareff constant \(\gamma_+''(A)=\max_{\text{HKZ}(\Lambda)}{\sqrt{A_1/A_n}}\)) where \(\min\) and \(\max\) are for all the \(\text{HKZ}\) (Hermite-Korkine-Zolotareff) reductions of a lattice \(\Lambda\) of rank \(n\) of an euclidean space \(E\) with \(A\) a Gram \(\text{HKZ}\)-reduced matrix of \(\Lambda\). These invariants are introduced to give an upper bound of the Bergé-Martinet constant \(\gamma^\prime(A)=\sqrt{\gamma(A)\gamma(A^{-1})}\) where \(\gamma(A)\) is the Hermite invariant and to give (conjecturally) a lower bound of \(\gamma(A)\) [\textit{A.M. Bergé}, J. Number Theory 52, No. 2, 284--298 (1995; Zbl 0829.11036) and \textit{J. Martinet}, Perfect lattices in Euclidean spaces. Grundlehren der Mathematischen Wissenschaften. 327. Berlin: Springer (2003; Zbl 1017.11031)]. This article introduces the \textit{antilaminations}, obtained by successive maximizations of \(A_n\), \(A_{n-1},\dots,A_2\) and the corresponding invariants \(\gamma_\pm'''(A)\), examines the relations between these two invariants to compute the value of these constants for the laminated lattices defined by Conway and Sloane. The author defines and examines the behaviour of recurrence of \(\gamma_{n,\pm}''\) and \(\gamma_{n,\pm}'''\) and derives some inequalities, formulates the conjecture \(\gamma_{5,-}''=\sqrt{2}\) and derives some consequences (e.g. \(\gamma_8=2\)) and he shows the unexpected discontinuity of these invar…
reviewed by: Roland Quême
Mathematics Subject Classification ID: 11H50
Mathematics Subject Classification ID: 11H06
zbMATH DE Number: 5774797
zbMATH Keywords: minima of quadratic forms
zbMATH Keywords: Hermite-Korkine-Zolotareff reduction
zbMATH Keywords: laminations and anti-laminations
zbMATH Keywords: Blichfeldt lattices
MaRDI profile type: Publication
OpenAlex ID: W2318847769
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DOI: 10.5802/JTNB.688
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description: scientific article; zbMATH DE number 5774797