Small solutions to a given quadratic form with a variable modulus
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.
scientific article; zbMATH DE number 166047