Editing Ehrhart polynomial (section)

==Ehrhart series==
We can define a [[generating function]] for the Ehrhart polynomial of an integral {{math|''d''}}-dimensional polytope {{math|''P''}} as

: <math> \operatorname{Ehr}_P(z) = \sum_{t\ge 0} L(P, t)z^t. </math>

This series can be expressed as a [[rational function]]. Specifically, Ehrhart proved (1962) that there exist complex numbers <math>h_j^*</math>, <math>0 \le j \le d</math>, such that the Ehrhart series of {{math|''P''}} is<ref name=ehrhart/>

:<math>\operatorname{Ehr}_P(z) = \frac{\sum_{j=0}^d h_j^\ast(P) z^j}{(1 - z)^{d + 1}}, \qquad \sum_{j=0}^d h_j^\ast(P) \neq 0.</math>

[[Richard P. Stanley]]'s non-negativity theorem states that under the given hypotheses, <math>h_j^*</math> will be non-negative integers, for <math>0 \le j \le d</math>.

Another result by Stanley shows that if {{math|''P''}} is a lattice polytope contained in {{math|''Q''}}, then <math>h_j^*(P) \le h_j^*(Q)</math> for all {{math|''j''}}.<ref>{{citation|last1=Stanley|first1=Richard|title=A monotonicity property of <math>h</math>-vectors and <math>h^*</math>-vectors|journal=[[European Journal of Combinatorics]]|year=1993|volume=14|issue=3 |pages=251–258 |doi=10.1006/eujc.1993.1028|doi-access=free}}</ref>  The <math>h^*</math>-vector is in general not unimodal, but it is whenever it is symmetric and the polytope has a regular unimodular triangulation.<ref>{{citation|last1=Athanasiadis|first1=Christos A.|title=''h''*-Vectors, Eulerian Polynomials and Stable Polytopes of Graphs| journal= [[Electronic Journal of Combinatorics]]| year=2004| volume=11| issue=2|doi=10.37236/1863| url= http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i2r6| doi-access=free}}</ref>

===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>