SMALL ZEROS OF QUADRATIC FORMS MOD P2

Let Q(x) be a quadratic form over ℤ in n variables, p be an odd prime and ∥x∥ = max i |x i |. A solution of the congruence Q(x) ≡ 0 (mod p 2 ) is said to be nontrivial if p ∤ x i for some i. We prove that if this congruence has a nontrivial solution, then it has a nontrivial solution with ∥x∥ ≤ p. We also give estimates on the number of small nontrivial solutions of the congruence and show that there exists a set of n linearly independent nontrivial solutions of size ∥x∥ ≤ (2 n+1 + 1)p, provided that n ≥ 4 is even and Q(x) is nonsingular (mod p).