Upper bounds on the magnitude of solutions of certain linear systems   with integer coefficients
  In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals.   Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some positive integer   $k$. We show that if the system has a nontrivial solution then there exists a nontrivial solution $\x=(x_1,...,x_n)\trans$ such that $\frac{|x_j|}{|x_i|}\le k^{n-1}$ for each $i,j$ satisfying $x_ix_j\ne 0$. This inequality is sharp.   We also prove a conjecture of A. Tyszka related to our results. 
Pedro J. Freitas, Shmuel Friedland and Gaspar Porta
math.CA