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Proceedings of the American Mathematical Society

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Bounds on positive integral solutions of linear Diophantine equations

Authors: I. Borosh and L. B. Treybig
Journal: Proc. Amer. Math. Soc. 55 (1976), 299-304
MathSciNet review: 0396605
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Abstract | References | Additional Information

Abstract: Assuming the existence of a solution, we find bounds for small solutions $ x$ of the finite matrix equation $ Ax = B$, where each entry of $ A,B$ is an integer, and $ x$ is a nontrivial column vector with nonnegative integer entries.

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  • [1] W. Haken, Theorie der Normalflächen, Acta Math. 105(1961), 245-375. MR25 #4519a. MR 0141106 (25:4519a)
  • [2] A. J. Goldman and A. W. Tucker, Polyhedral convex cones, Linear Inequalities and Related Systems, Ann. of Math. Studies, no. 38, Princeton Univ. Press, Princeton, N.J., 1956, pp. 19-40. MR19, 446. MR 0087974 (19:446g)
  • [3] O. L. Mangasarian, Nonlinear programming, McGraw-Hill, New York, 1969, p. 31. MR40 #5263. MR 0252038 (40:5263)
  • [4] I. Niven and H. S. Zuckerman, An introduction to the theory of numbers, 2nd ed., Wiley, New York, 1966, pp. 107-109. MR33 #3981. MR 0195783 (33:3981)
  • [5] H. Schubert, Bestimmung der Primfaktorzerlegung von Verkettungen, Math. Z. 76 (1961), 116-148. MR25 #4519b. MR 0141107 (25:4519b)
  • [6] L. B. Treybig, Bounds in piecewise linear topology, Trans. Amer. Math. Soc. 201(1975), 385-405. MR 0350746 (50:3238)

Additional Information

Keywords: Linear equations, integer entries, nonnegative integral solution, bounds, polyhedral convex cones, minors, adjoint matrix
Article copyright: © Copyright 1976 American Mathematical Society

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