On small zeros of quadratic forms over finite fields (Q1119677)

\textit{D. R. Heath-Brown} proved [Glasg. Math. J. 27, 87--93 (1985; Zbl 0581.10008)] that a quadratic form \(Q\) over \({\mathbb Z}/p{\mathbb Z}\) in \(m\geq 4\) variables has a nontrivial zero \(x=(x_ 1,\dots, x_ m)\) \((0\leq x_ i<p)\) with \(\max (x_ i)\leq c p^{1/2} \log p\) (\(c\) an absolute constant). A generalization to finite fields has been given by \textit{J. Sander} [Small solutions of quadratic equations over finite fields (German), Diss. Hannover (1987; Zbl 0628.10022)]. In the present paper, the author proves (apparently independently) the same result as Sander by similar methods.