Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
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
    

1 Answer 1

up vote 1 down vote accepted

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.

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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.