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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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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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    $\begingroup$ I think you mean $2^k$. $\endgroup$
    – David E Speyer
    Commented Aug 18, 2010 at 18:21
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    $\begingroup$ 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. $\endgroup$
    – David E Speyer
    Commented Aug 18, 2010 at 18:23
  • $\begingroup$ Robin or David, Could you give a reference? I did mean Z^d. Thanks! $\endgroup$
    – user6976
    Commented Aug 18, 2010 at 23:23
  • $\begingroup$ The formula is proved here: math.sfsu.edu/beck/papers/noprint.pdf $\endgroup$
    – user6976
    Commented Aug 19, 2010 at 2:35
  • $\begingroup$ Thanks for the answer, Robin! It was needed here: front.math.ucdavis.edu/1008.3868 $\endgroup$
    – user6976
    Commented Sep 2, 2010 at 0:08

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