Small zeros of quadratic forms modulo \(p\).
type: Item
instance of: scholarly article
title: Small zeros of quadratic forms modulo \(p\).
zbMATH Open document ID: 0684.10017
DOI: 10.1016/0022-314X(89)90065-6
author: Todd Cochrane
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: Let \(Q(x)\) be a quadratic form in \(n\) variables, and \(p\) a prime not dividing \(\det Q\). Write \(Q^{-1}(x)\) for the form whose matrix is inverse to that of \(Q\), modulo \(p\). It is shown that, for even \(n\), there is a non-zero integer vector \(x\) with \(| x| \ll p^{1/2}\), for which either \(p | Q(x)\) or \(p | Q^{-1}(x)\). It is conjectured that, for \(n\geq 4\) one can find such an \(x\) with \(p | Q(x)\), and this is proved whenever \(Q\) is a sum of two forms in disjoint sets of \(r\) and \(n-r\) variables with \(2\leq r\leq n-2\). Finally, when \(n>3+4 \log p/\log 2\), it is shown that \(p | Q(x)\) has a non-zero solution with \(| x| <p^{1/2}\). The proofs use two quite different methods, due to the reviewer [Glasg. Math. J. 27, 87--93 (1985; Zbl 0581.10008)] and \textit{A. Schinzel}, \textit{H. P. Schlickewei} and \textit{W. M. Schmidt} [Acta Arith. 37, 241--248 (1980; Zbl 0446.10026)], respectively.
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11D79
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11H55
zbMATH DE Number: 4120247
zbMATH Keywords: small zeros
zbMATH Keywords: congruences in many variables
zbMATH Keywords: quadratic congruences
zbMATH Keywords: quadratic form
MaRDI profile type: Publication
cites work: Weighted Quadratic Partitions Over a Finite Field
cites work: Small solutions of congruences over algebraic number fields
cites work: Small solutions of quadratic congruences
cites work: Small solutions of quadratic congruences and small fractional parts of quadratic forms
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description: scientific article; zbMATH DE number 4120247