Primitive zeros of quadratic forms mod $p^2$

Abstract

Let $Q({\bf{x}}) = Q(x_1 ,x_2 ,\ldots,x_n )$ be a quadratic form with integer coefficients, $p$ be an odd prime and $\left\| \bf{x} \right\| = \max _i \left| {x_i } \right|.$ A solution of the congruence $Q({\mathbf{x}}) \equiv {\mathbf{0}}\;(\bmod\; p^2 )$ is said to be a primitive solution if $p\nmid x_i $ for some $i$. In this paper, we seek to obtain primitive solutions of this congruence in small rectangular boxes of the type $ \mathcal{B} = \{ {\mathbf{x}} \in \mathbb{Z}^n : |x_i| \le M_i ,\;1 \leqslant i \leqslant n\} $ where for $1 \le i \le l$ we have $M_i \le p$, while for $i>l$ we have $M_i>p$. In particular, we show that if $n \ge 4$, $n$ even, $l \le \frac n2-2$, and $Q$ is nonsingular $\pmod p$, then there exists a primitive solution with $x_i = 0$, $1 \le i \le l$, and $|x_i| \le 2^{\frac {4n+3}{n-l}} p^{\frac n{n-l}} +1$, for $l<i \le n$.