Small solutions to a given quadratic form with a variable modulus
type: Item
instance of: scholarly article
title: Small solutions to a given quadratic form with a variable modulus
zbMATH Open document ID: 0770.11021
DOI: 10.1016/0022-314X(92)90108-2
author: David Grant
published in: Journal of Number Theory
publication date: 1993-05-16T00:00:00Z
review text: Let \(Q\) be an integral quadratic form in 4 or more variables. It is conjectured that there is an absolute constant \(c\), such that for any modulus \(m\) there is a nontrivial solution of \(Q(x_ 1,\dots,x_ n)\equiv 0\pmod m\), having \(\| {\mathbf x}\|\leq c\sqrt m\). This was proved by \textit{T. Cochrane} [J. Number Theory 37, 92-99 (1991; Zbl 0713.11031)] for prime values of \(m\), but the general case remains in doubt. The present paper shows that the required estimate holds for positive definite forms, providing one allows \(c\) to depend on \(Q\). The proof, which is quite straightforward, uses modular forms. It seems to the reviewer that the result is also an easy consequence of a theorem of \textit{B. J. Birch} and \textit{H. Davenport} [Proc. Camb. Philos. Soc. 54, 135-138 (1958; Zbl 0081.044)], from which it follows that if \(\sum^ n_{i=1}a_ ix^ 2_ i=0\) has a nontrivial solution then there is such a solution with \(\sum^ n_{i=1}| a_ i| x^ 2_ i\ll_ n\prod| a_ i|\). Since we can clearly diagonalize \(Q\) in the present problem, the above result may be applied to the form \(Q({\mathbf x})-mx^ 2_{n+1}\), to obtain the required estimate. Indeed one sees that \(Q\) need not be positive definite.
Mathematics Subject Classification ID: 11D79
Mathematics Subject Classification ID: 11E45
zbMATH DE Number: 166047
zbMATH Keywords: congruence
zbMATH Keywords: small solutions
zbMATH Keywords: integral quadratic form
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
cites work: Q3030890
cites work: Small zeros of quadratic congruences modulo pq
cites work: Small zeros of quadratic forms modulo \(p\). III
cites work: Small solutions of quadratic congruences
cites work: Small solutions of quadratic congruences and small fractional parts of quadratic forms
full work available at URL: 0022-314x(92)90108-2
OpenAlex ID: W1994527802
description: scientific article; zbMATH DE number 166047