Small Solutions of Quadratic Diophantine Equations

Abstract

We consider quadratic diophantine equations of the shape

$$ Q(x_1, \ldots, x_s) = 0 \qquad (1)$$

for a polynomial $Q(X_1, \ldots, X_s) \in \mathbf{Z}[X_1, \ldots, X_s]$ of degree $2$. Let $H$ be an upper bound for the absolute values of the coefficients of $Q$, and assume that the homogeneous quadratic part of $Q$ is non-singular. We prove, for all $s \ge 3$, the existence of a polynomial bound $\Lambda_s(H)$ with the following property: if equation (1) has a solution $\mathbf{x} \in \mathbf{Z}^s$ at all, then it has one satisfying

$$ |x_i| \le \Lambda_s(H) \quad (1 \le i \le s). $$

For $s = 3$ and $s = 4$ no polynomial bounds $\Lambda_s(H)$ were previously known, and for $s \ge 5$ we have been able to improve existing bounds quite significantly.