Quadratic forms which have only large zeros
type: Item
instance of: scholarly article
title: Quadratic forms which have only large zeros
zbMATH Open document ID: 0684.10016
DOI: 10.1007/BF01318806
author: Hans Peter Schlickewei
author: Wolfgang M. Schmidt
published in: Monatshefte für Mathematik
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178383
review text: Let Q(\({\mathfrak x})=\sum_{i,j}Q_{ij}x_ ix_ j\) be a symmetric quadratic form in n variables with rational integer coefficients. Cassels proved that if Q has a non-trivial integral zero, then it has one satisfying \(| {\mathfrak x}| \ll | Q|^{(n-1)/2}.\) Here \(| {\mathfrak x}|\) and \(| Q|\) are the maximum modulus of the coordinates (respectively coefficients) of \({\mathfrak x}\) and Q, and the \(\ll\) constants will always depend only on n. Kneser showed that Cassel's result is best possible. In earlier papers the authors showed that if Q is non-singular and vanishes on a d-dimensional rational subspace for some \(n\geq 2d>0\), then Q has a non-trivial integral zero satisfying \(| {\mathfrak x}| \ll | Q|^{(n-d)/2d}.\) The first theorem in this paper constructs quadratic forms which show that this higher dimensional analogue of Cassel's theorem is best possible. Next, the authors consider the question of finding n linearly independent integral zeros of Q. Earlier they proved that if Q is non-singular and vanishes on a d-dimensional rational subspace with \(n\geq 2d>0\), then there are n linearly independent integral zeros \({\mathfrak x}_ 1,...,{\mathfrak x}_ n\) of Q satisfying \[ (*)\quad | {\mathfrak x}_ 1| \cdot.... \cdot | {\mathfrak x}_ n| \quad \ll \quad | Q|^{(n^ 2/2d)-d}. \] The second theorem in this paper gives a construction which shows that (*) is best possible. In particular, Davenport had conjectured that (*) should hold with exponent \((n^ 2/2d)-(n/2)\); the authors' estimate shows that Davenport's conjecture is only true in the case \(n=2d\).
Mathematics Subject Classification ID: 11D04
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11H55
zbMATH DE Number: 4120246
zbMATH Keywords: lower bounds for zeros
zbMATH Keywords: symmetric quadratic form
zbMATH Keywords: higher dimensional analogue of Cassel's theorem
zbMATH Keywords: Davenport's conjecture
MaRDI profile type: Publication
cites work: Q5848294
cites work: Q3234241
cites work: Homogeneous quadratic equations
cites work: Kleine Nullstellen homogener quadratischer Gleichungen. (Small zeros of homogeneous quadratic equations)
cites work: Quadratic Geometry of Numbers
cites work: Small Zeros of Quadratic Forms
OpenAlex ID: W2088095481
description: scientific article; zbMATH DE number 4120246