On a conjecture of Schmutz
type: Item
instance of: scholarly article
title: On a conjecture of Schmutz
zbMATH Open document ID: 1163.11028
author: Thomas A. Willging
published in: Archiv der Mathematik
publication date: 2009-01-30T00:00:00Z
review text: Motivated by related observations from the hyperbolic world, Paul Schmutz conjectured in the mid nineties of last century that among all Euclidean lattices in dimension three and with covolume 1 there is one lattice which is extremal in the sense that for every natural number \(k\) its \(k\)-th length (counted without multiplicities) is strictly larger than the \(k\)-th length of any other lattice. By asymptotic considerations, this would have to be the normalized root lattice \(A_2.\) The author gives an example of a lattice violating this conjecture and suggests that this phenomenon is the only counterexample to be found. This suggestion in particular holds for the lattice giving the counterexample to Schmutz' original conjecture. It is furthered by yet unpublished results of Willging.
reviewed by: Stefan Kühnlein
Mathematics Subject Classification ID: 11E20
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 5500765
zbMATH Keywords: length spectrum
zbMATH Keywords: root lattice
zbMATH Keywords: quadratic forms
Wikidata QID: Q123165870
MaRDI profile type: Publication
full work available at URL: s00013-008-2753-2
OpenAlex ID: W2049917142
DOI: 10.1007/S00013-008-2753-2
description: scientific article; zbMATH DE number 5500765