Editing Ehrhart polynomial (section)

===Ehrhart series for rational polytopes===
As in the case of polytopes with integer vertices, one defines the Ehrhart series for a rational polytope. For a ''d''-dimensional rational polytope {{math|''P''}}, where {{math|''D''}} is the smallest integer such that {{math|''DP''}} is an integer polytope ({{math|''D''}} is called the denominator of {{math|''P''}}), then one has

:<math>\operatorname{Ehr}_P(z) = \sum_{t\ge 0} L(P, t)z^t = \frac{\sum_{j=0}^{D(d+1)} h_j^\ast(P) z^j}{\left(1 - z^D\right)^{d + 1}},</math>

where the <math>h_j^*</math> are still non-negative integers.<ref>{{citation|last=Stanley|first=Richard P.|authorlink=Richard P. Stanley|title=Decompositions of rational convex polytopes|journal=Annals of Discrete Mathematics|date=1980|volume=6|pages=333–342| doi=10.1016/s0167-5060(08)70717-9|isbn=9780444860484}}</ref><ref>{{citation| last1=Beck| first1=Matthias| last2=Sottile| first2= Frank|title=Irrational proofs for three theorems of Stanley|journal=[[European Journal of Combinatorics]]|date=January 2007| volume =28|issue=1|pages=403–409|doi=10.1016/j.ejc.2005.06.003|arxiv=math/0501359| s2cid=7801569}}</ref>