Positive values of inhomogeneous indefinite ternary quadratic forms of type \((2,1)\)
type: Item
instance of: scholarly article
title: Positive values of inhomogeneous indefinite ternary quadratic forms of type \((2,1)\)
zbMATH Open document ID: 0859.11042
author: Madhu Raka
author: Urmila Rani
published in: Hokkaido Mathematical Journal
publication date: 1997-04-09T00:00:00Z
review text: Let \(Q(X)= Q(x_1, \dots, x_n)\) be a real indefinite \(n\)-ary quadratic form of determinant \(D\neq 0\) and of type \((r,n-r)\). \textit{H. Blaney} [J. Lond. Math. Soc. 23, 153-160 (1948; Zbl 0031.20404)] proved that there exist real numbers \(\Gamma\), depending only on \(n\) and \(r\), such that for any \(C\in\mathbb{R}^n\) there exists \(X\in\mathbb{Z}^n\) such that \(0<Q(X+C) \leq (\Gamma |D |)^{1/n}\). Then the problem arose of determining the minimum \(\Gamma_{r,n-r}\) and the \(k\)th successive inhomogeneous minimum \(\Gamma^{(k)}_{r,n-r}\) of all such numbers \(\Gamma\) for the forms \(Q(X)\). Further investigations gave the values of these minima for various \(n,r\) and \(k\). The present paper considers ternary forms of type (2,1) and proves that \(\Gamma^{(2)}_{2,1} = 8/3\). All the critical forms are given. Earlier, for zero ternary forms of the mentioned type the first author [J. Aust. Math. Soc., Ser. A 55, 334-354 (1993; Zbl 0798.11028)] obtained the first four minima.
reviewed by: Guram Gogishvili
Mathematics Subject Classification ID: 11H50
Mathematics Subject Classification ID: 11E20
zbMATH DE Number: 910516
zbMATH Keywords: inhomogeneous indefinite ternary quadratic forms
zbMATH Keywords: lattice
zbMATH Keywords: continued fractions
zbMATH Keywords: successive inhomogeneous minima
zbMATH Keywords: successive inhomogeneous minimum
zbMATH Keywords: critical forms
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1351516723
OpenAlex ID: W1974816166
DOI: 10.14492/HOKMJ/1351516723
description: scientific article; zbMATH DE number 910516