Editing Ehrhart polynomial (section)

==Examples==
[[File:Second dilate of a unit square.png|thumbnail|This is the second dilate, <math>t = 2</math>, of a unit square. It has nine integer points.]]

Let {{math|''P''}} be a {{math|''d''}}-dimensional [[unit cube|unit]] [[hypercube]] whose vertices are the integer lattice points all of whose coordinates are 0 or 1. In terms of inequalities,

:<math> P = \left\{x\in\R^d : 0 \le x_i \le 1; 1 \le i \le d\right\}.</math>

Then the {{math|''t''}}-fold dilation of {{math|''P''}} is a cube with side length {{math|''t''}}, containing {{math|(''t'' + 1)<sup>''d''</sup>}} integer points. That is, the Ehrhart polynomial of the hypercube is {{math|''L''(''P'',''t'') {{=}} (''t'' + 1)<sup>''d''</sup>}}.<ref>{{citation|title=Triangulations: Structures for Algorithms and Applications|volume=25|series=Algorithms and Computation in Mathematics|first1=Jesús A.|last1=De Loera|author1-link=Jesús A. De Loera|first2=Jörg|last2=Rambau|first3=Francisco|last3=Santos|author3-link= Francisco Santos Leal 
|publisher=Springer|year=2010|isbn=978-3-642-12970-4
|contribution=Ehrhart polynomials and unimodular triangulations
|page=475
|url=https://books.google.com/books?id=SxY1Xrr12DwC&pg=PA475}}</ref><ref>{{citation|first1=Richard J.|last1=Mathar|arxiv=1002.3844
|title=Point counts of <math>D_k</math> and some <math>A_k</math> and <math>E_k</math> integer lattices inside hypercubes
|year=2010
|bibcode=2010arXiv1002.3844M
}}</ref> Additionally, if we evaluate {{math|''L''(''P'', ''t'')}} at negative integers, then

: <math>L(P, -t) = (-1)^d (t - 1)^d = (-1)^d L(\operatorname{int}(P), t),</math>

as we would expect from Ehrhart–Macdonald reciprocity.

Many other [[figurate number]]s can be expressed as Ehrhart polynomials. For instance, the [[square pyramidal number]]s are given by the Ehrhart polynomials of a [[square pyramid]] with an integer unit square as its base and with height one; the Ehrhart polynomial in this case is {{math|{{sfrac|1|6}}(''t'' + 1)(''t'' + 2)(2''t'' + 3)}}.<ref>{{citation
 | last1 = Beck | first1 = Matthias
 | last2 = De Loera | first2 = Jesús A. | author2-link = Jesús A. De Loera
 | last3 = Develin | first3 = Mike | author3-link = Mike Develin
 | last4 = Pfeifle | first4 = Julian
 | last5 = Stanley | first5 = Richard P. | author5-link = Richard P. Stanley
 | contribution = Coefficients and roots of Ehrhart polynomials
 | location = Providence, RI
 | mr = 2134759
 | pages = 15–36
 | publisher = [[American Mathematical Society]]
 | series = Contemporary Mathematics
 | title = Integer Points in Polyhedra—Geometry, Number Theory, Algebra, Optimization
 | volume = 374
 | year = 2005}}</ref>