# Estimate the number of integral solutions inside a convex polyhedron

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