Estimates on conjectures of Minkowski and Woods
type: Item
instance of: scholarly article
title: Estimates on conjectures of Minkowski and Woods
zbMATH Open document ID: 1228.11096
author: R. J. Hans-Gill
author: Madhu Raka
author: Ranjeet Sehmi
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2010-10-01T00:00:00Z
review text: Let \(L_i=a_{i1}x_1 + \dots + a_{in}x_n\), \(1\leq i \leq n\), be \(n\) real linear forms in \(n\) real variables with determinant \(\Delta=\det(a_{ij}) \neq 0\). Let \(c_1, \ldots, c_n\) be real numbers. \textit{H. Minkowski} is believed to have conjectured that there exist integers \(u_1, \ldots, u_n\) such that \[ \prod_{i=1}^n |L_i(u_1, \ldots, u_n)+c_i| \leq \frac{1}{2^n} |\Delta|. \] He proved this for \(n=2\), [Math. Ann. 54, 91--124 (1901; JFM 31.0213.02)] and it has now been verified for \(2\leq n\leq 8\). Define \[ M_n = \sup_{L_1, \ldots, L_n} \sup_{(c_1, \ldots, c_n) \in \mathbb R^n} \inf_{(u_1, \ldots, u_n) \in \mathbb Z^n} \prod_{1\leq i \leq n} |L_i(u_1,\ldots, u_n)+c_i|. \] Minkowski's Conjecture is equivalent to \(M_n \leq \frac{1}{2^n} |\Delta|\). \textit{N. Chebotarev} [Vierteljahresschr. Naturforsch. Ges. 85, Beibl. Nr. 32, 27--30 (Rudolf Fueter Festschr.) (1940; Zbl 0023.20701 and JFM 66.0185.01)] proved the weaker inequality \(M_n \leq \frac{1}{2^{n/2}} |\Delta|\). Several authors, including \textit{I. V. Il'in} [Algebraic systems. Ivanovo: Ivanov. Gos. Univ., 115--125 (1991; Zbl 0956.00528)] have given improved bounds for \(M_n\) of the form \(M_n \leq \frac{1}{\nu_n 2^{n/2}} |\Delta|\). In this paper, the authors give bounds for \(\nu_n\) that improve Il'in's bounds in the range \(9\leq n \leq 22\). For example, Il'in had \(\nu_9=3.315\ldots\); the authors improve this to \(\nu_9=19.396\ldots\). The authors also consider a related conjecture due to \textit{A. C. Woods} [Mathematika 12, 143--150 (1965; Zbl 0135.10701)]. Woods's conjecture is also known to be true for \(n\leq 8\). In this paper, the authors obtain an approximation to Wood's conjecture that is valid for \(n\geq 9\), and they use this to obtain the above mentioned improvements for \(\nu_n\).
Mathematics Subject Classification ID: 11H46
zbMATH DE Number: 5794230
zbMATH Keywords: geometry of numbers
zbMATH Keywords: Minkowski's conjecture
Wikidata QID: Q123012191
reviewed by: Sidney W. Graham
MaRDI profile type: Publication
full work available at URL: s13226-010-0034-9
OpenAlex ID: W2055241707
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DOI: 10.1007/S13226-010-0034-9
description: scientific article; zbMATH DE number 5794230