Small zeros of quadratic forms mod $P^2$

Abstract:

Let $Q(\mathbf x)$ be a quadratic form over $\mathbb Z$ in $n$ variables, $p$ be an odd prime and $\| \mathbf x\|= \max _i |x_i|$. A solution of the congruence $Q(\mathbf x) \equiv 0 \pmod {p^2}$ is said to be nontrivial if $p \nmid x_i$ for some $i$. We prove that if this congruence has a nontrivial solution, then it has a nontrivial solution with $\|\mathbf x\|\le p$. 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 $\|\mathbf x\| \le (2^{n+1}+1)p$, provided that $n \ge 4$ is even and $Q(\mathbf x)$ is nonsingular $\pmod p$.