Values of non-homogeneous indefinite quadratic forms
type: Item
instance of: scholarly article
title: Values of non-homogeneous indefinite quadratic forms
zbMATH Open document ID: 0803.11033
author: V. C. Dumir
author: R. J. Hans-Gill
author: A. C. Woods
published in: Journal of Number Theory
publication date: 1995-01-03T00:00:00Z
review text: Let \(f(X)\) be a real indefinite quadratic form in \(n\) variables of determinant \({\mathcal D}\neq 0\) and signature \(s\). It is well known that there exists a real number \(c>0\) depending only on \(n\) and \(s\) such that for any \(C\in \mathbb{R}^ n\) there exists \(X\in \mathbb{Z}\) such that \[ | f(X+C) |\leq (c | {\mathcal D}| )^{1/n}. \tag{1} \] The problem is to determine the infimum \(C_{n,s}\) of all such numbers \(c\). The first investigations gave the values of \(C_{n,s}\) only in some special cases. Then \textit{G. L. Watson} [Proc. Lond. Math. Soc., III. Ser. 12, 564-576 (1962; Zbl 0107.041)] gave a formula for \(C_{n,s}\) if \(n\geq 21\), \(s\) is arbitrary, and conjectured that the result is true for all \(n\geq 4\). Further investigations of the problem are finally completed in the present paper. So Watson's conjecture is established. If in the discussed problem the inequality (1) is changed to \(0< f(X+C)< (\Omega | {\mathcal D}|)^{1/n}\) with real \(\Omega> 0\), then the problem of determining the infimum \(\Omega_{n,s}\) of all such numbers \(\Omega\) arises. The authors prove that the values of \(\Omega_{n,s}\) for a fixed \(n\) are invariant under the signature modulo 8, what was conjectured by Bambah, Dumir and Hans-Gill. Now \(\Omega_{n,s}\) are determined for all meaningful pairs \(n\), \(s\) with the exception of \(\Omega_{6,-2}\) and \(\Omega_{5, -3}\). The proofs of both conjectures follow from the results of Margulis and Watson on rational forms.
Mathematics Subject Classification ID: 11H50
Mathematics Subject Classification ID: 11D85
zbMATH DE Number: 584039
zbMATH Keywords: congruentially equivalent forms
zbMATH Keywords: representations of integers by forms
zbMATH Keywords: real indefinite quadratic form
reviewed by: Guram Gogishvili
MaRDI profile type: Publication
full work available at URL: jnth.1994.1033
OpenAlex ID: W2067809116
DOI: 10.1006/JNTH.1994.1033
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description: scientific article; zbMATH DE number 584039