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



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


