Small zeros of quadratic forms modulo \(p\). (Q1825226)

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.