Isotropic subspaces of rational quadratic forms
type: Item
instance of: scholarly article
title: Isotropic subspaces of rational quadratic forms
zbMATH Open document ID: 0645.10020
DOI: 10.1007/BF01160677
author: Hans Peter Schlickewei
author: Wolfgang M. Schmidt
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174047
review text: Let \(\mathcal F = \mathcal F(x_1,\ldots,x_n)\) be a nondegenerate quadratic form with rational integer coefficients \(f_{ij} = f_{ji}\). Suppose \(\mathcal F\) vanishes on a \(d\)-dimensional rational linear subspace. We give a qualitative and quantitative description of such subspaces. A particular result is as follows: Suppose \(n=2d\). Then there exist linearly independent integral zeros \(\mathfrak x_1,\ldots,\mathfrak x_n\) of \(\mathcal F\) satisfying \[ | \mathfrak x_1| \cdots | \mathfrak x_ n| \ll F^{n/2}, \] where \(F\) is the maximum modulus of the coefficients of \(\mathcal F\) and where for \(\mathfrak x=(x_1,\ldots,x_n)\) we put \(| \mathfrak x| =\max _{1\leq i\leq n}| x_i|\).
Mathematics Subject Classification ID: 11H55
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E16
zbMATH DE Number: 4051724
zbMATH Keywords: quadratic form
zbMATH Keywords: rational linear subspace
zbMATH Keywords: integral zeros
MaRDI profile type: Publication
cites work: Q3249039
cites work: Q5848294
cites work: Q4177654
cites work: Homogeneous quadratic equations
cites work: Q5814388
cites work: Kleine Nullstellen homogener quadratischer Gleichungen. (Small zeros of homogeneous quadratic equations)
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cites work: Small linearly independent zeros of quadratic forms
OpenAlex ID: W1968400160
description: scientific article; zbMATH DE number 4051724