Small solutions of linear diophantine equations (Q1074620)

Given a system of linear diophantine equations and let (*) \(Ax=B\) be its matrix form, where \(A=(a_{ij})\) is a \(m\times n\), \(x=(x_ k)\) and \(B=(a_{k,n+1})\) are \(m\times 1\) matrices. Further, given integers \(1\leq j_ 1<...<j_ m\leq n+1\), let \(d_{j_ 1,...,j_ m}=\det (a_{i,j_ r}),\) \(1\leq i,r\leq m\), \(X=\sup \{| d_{j_ 1,...,j_ m}|:\quad j_ s\leq n\},\) \(Y=\sup \{| d_{j_ 1,...,j_ m}| \}.\) It is proved, that if the rows of A are linearly independent and (*) has a nonzero non-negative integral solution, then there exist a nonzero integral solution \(x=(x_ i)\) of (*) and integers \(1\leq j_ 1<...<j_ m\leq n\) such that \(d_{j_ 1,...,j_ m}\neq 0\), \(0\leq x_ p\leq X\) for \(p\not\in \{j_ 1,...,j_ m\}\) and \(0\leq x_ p\leq (n-m)X+Y\) otherwise. The paper also gives other estimates of small solutions to a system of linear equations.