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

Abstract

Let\(Q(\underline{\underline x} ) = Q(x_1 , \cdot \cdot \cdot x_n )\) be a quadratic form with integer coefficients and letp denote a prime. Cochrane[1] proved that ifn≥4 then\(Q(\underline{\underline x} ) = 0(\bmod p)\) has a solution\(\underline{\underline x} \ne \underline{\underline 0} \) satisfying\(\left| {\underline{\underline x} } \right| \ll \sqrt p \), where\(\left| {\underline{\underline x} } \right| = \max \left| {x_i } \right|\). The aim of the present paper is to generalize the above result to finite fields.