Small zeros of quadratic forms modulo \(p\). III (Q749576)

[For parts I, II cf. ibid. 33, 286--292 (1989; Zbl 0684.10017), Analytic number theory, Proc. Conf. in Honor of Paul T. Bateman, Urbana/IL 1989, Prog. Math. 85, 91--94 (1990; Zbl 0715.11015).] Let \(Q(x_ 1,\dots, x_ 4)\) be an integral quadratic form. It was shown by the reviewer [Glasg. Math. J. 27, 87--93 (1985; Zbl 0581.10008)], that the congruence \(Q\equiv 0\pmod p\) has a non-zero solution, for any prime modulus \(p\), with \(\| x\| \ll p^{1/2} \log p\). The present paper removes the extraneous factor \(\log p\), by using a superior version of the reviewer's key lemma. The result is now best possible, apart from the order constant. It would be nice to extend the result to composite moduli.