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How can I compute an estimate of the number of integral solutions (points) inside a bounded convex polyhedron with dimension $d$? I'm interested more in an efficient way to estimate the number of integral solutions than in a very close estimate.

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The $d$-volume is a reasonable estimate. This is explicit for $d=2$ by Pick's theorem. –  Will Jagy May 6 '13 at 21:26
@Maesumi: why the factor $2^d$? –  Ross Millikan May 6 '13 at 21:28

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By integral solutions do you mean lattice points? That is, points with all coordinates integral? The first estimate would be just the volume of the polyhedron. It will be more accurate the larger the polyhedron is. The error is bounded by the volume within one unit of the surface, so by the surface area of the polyhedron (I believe not twice as points that come in one side leave on the other). As the minimum dimension of the polyhedron rises, the relative error will shrink.

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Yes, lattice points. Sorry, I'm an engineer, so my use of mathematical terminology is sometimes imprecise. :) –  synaptik May 6 '13 at 21:28

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