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Abstract: This paper investigates properties of the minimal integral solutions of a linear diophantine equation. We present best possible inequalities that must be satisfied by these elements which improves on former results. We also show that the elements of the minimal Hilbert basis of the dual cone of all minimal integral solutions of a linear diophantine equation yield best approximations of a rational vector "from above". Relations between these cones are applied to the knapsack problem. (Update)
Context of citations to this paper: More
...of at most n vectors from the Hilbert basis. This also holds for cones arising from perfect graphs [18] and a class of cones described in [46]. However, in general at least n b1=6 nc elements of the Hilbert basis are needed to represent any integral vector in the cone....
...of at most n vectors from the Hilbert basis. This also holds for cones arising from perfect graphs [18] and a class of cones described in [46]. However, in general at least n #1 6n# elements of the Hilbert basis are needed to represent any integral vector in the cone....
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BibTeX entry: (Update)
M. Henk and R. Weismantel (1997), Hilbert bases of cones related to simultaneous Diophantine approximations and linear Diophantine equations, Konrad-Zuse-Zentrum fur Informationstechnik Berlin, Preprint SC 97-29. http://citeseer.ist.psu.edu/henk97hilbert.html More
@misc{ henk97hilbert,
author = "M. Henk and R. Weismantel",
title = "Hilbert bases of cones related to simultaneous Diophantine approximations
and linear Diophantine equations",
text = "M. Henk and R. Weismantel (1997), Hilbert bases of cones related to simultaneous
Diophantine approximations and linear Diophantine equations, Konrad-Zuse-Zentrum
fur Informationstechnik Berlin, Preprint SC 97-29.",
year = "1997",
url = "citeseer.ist.psu.edu/henk97hilbert.html" }
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