A sharp bound for solutions of linear Diophantine equations

Abstract

Let $Ax = b$ be an $m \times n$ system of linear equations with rank $m$ and integer coefficients. Denote by $Y$ the maximum of the absolute values of the $m \times m$ minors of the augmented matrix $\left ( {A,b} \right )$. It is proved that if the system has an integral solution, then it has an integral solution $x = \left ( {{x_i}} \right )$ with $\max \left | {{x_i}} \right | \leq Y$. The bound is sharp.