Geodesic
Bounds on Positive Integral Solutions of Linear Diophantine Equations II
Canadian mathematical bulletin, Tome 22 (1979) no. 3, pp. 357-361

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Let A be an m × n matrix of rank r and B an m × 1 matrix, both with integer entries. Let M2 be the maximum of the absolute values of the r × r minors of the augmented matrix (A | B). Suppose that the system A x = B has a non-trivial solution in non-negative integers. We prove (1) If r = n - 1 then the system A x = B has a non-negative non-trivial solution with entries bounded by M2. (2) If A has a r x n submatrix such that none of its r x r minors is 0 and x ≥ 0 is a solution of Ax=B in integers such that is minimal, then . 
Borosh, I.; Treybig, L. B. Bounds on Positive Integral Solutions of Linear Diophantine Equations II. Canadian mathematical bulletin, Tome 22 (1979) no. 3, pp. 357-361. doi: 10.4153/CMB-1979-045-2
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     author = {Borosh, I. and Treybig, L. B.},
     title = {Bounds on {Positive} {Integral} {Solutions} of {Linear} {Diophantine} {Equations} {II}},
     journal = {Canadian mathematical bulletin},
     pages = {357--361},
     year = {1979},
     volume = {22},
     number = {3},
     doi = {10.4153/CMB-1979-045-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1979-045-2/}
}
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