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What is the Ehrhart polynomial of the regular cross-polytope of dimension d? Are there published upper and lower estimates?

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up vote 11 down vote accepted

If you mean the polytope with vertices $(0,\ldots,0,\pm1,0,\ldots,0)$ then it is easily seen to be $$\sum_{k=0}^d 2^k{d\choose k}{x\choose k}.$$

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I think you mean $2^k$. –  David Speyer Aug 18 '10 at 18:21
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By the way, you are assuming that Mark wants to work with the lattice $\mathbb{Z}^d$. The other reasonable choice would be to work with the index two sublattice where the sum of the coordinates is odd. (And, thus, there are no interior lattice points in the polytope.) That should be some simple correction, but I'm not sure what. –  David Speyer Aug 18 '10 at 18:23
    
Thanks David –  Robin Chapman Aug 18 '10 at 18:59
    
Robin or David, Could you give a reference? I did mean Z^d. Thanks! –  Mark Sapir Aug 18 '10 at 23:23
    
The formula is proved here: math.sfsu.edu/beck/papers/noprint.pdf –  Mark Sapir Aug 19 '10 at 2:35

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