Editing Ehrhart polynomial (section)

==Definition==
Informally, if {{math|''P''}} is a [[polytope]], and {{math|''tP''}} is the polytope formed by expanding {{math|''P''}} by a factor of {{math|''t''}} in each dimension, then {{math|''L''(''P'', ''t'')}} is the number of [[integer lattice]] points in {{math|''tP''}}.

More formally, consider a [[lattice (group)|lattice]] <math>\mathcal{L}</math> in [[Euclidean space]] <math>\R^n</math> and a {{math|''d''}}-[[dimension]]al polytope {{math|''P''}} in <math>\R^n</math> with the property that all vertices of the polytope are points of the lattice. (A common example is <math>\mathcal{L} = \Z^n</math> and a polytope for which all vertices have [[integer]] coordinates.) For any positive integer {{math|''t''}}, let {{math|''tP''}} be the {{math|''t''}}-fold dilation of {{math|''P''}} (the polytope formed by multiplying each vertex coordinate, in a basis for the lattice, by a factor of {{math|''t''}}), and let

:<math>L(P,t) = \#\left(tP \cap \mathcal{L}\right)</math>

be the number of lattice points contained in the polytope {{math|''tP''}}. Ehrhart showed in 1962 that {{math|''L''}} is a rational [[polynomial]] of degree {{math|''d''}} in {{math|''t''}}, i.e. there exist [[rational number]]s <math>L_0(P),\dots,L_d(P)</math> such that:

:<math>L(P, t) = L_d(P) t^d + L_{d-1}(P) t^{d-1} + \cdots + L_0(P)</math>

for all positive integers {{math|''t''}}.<ref name=ehrhart>{{citation
 | last = Ehrhart | first = Eugène | authorlink=Eugène Ehrhart
 | journal = [[Comptes rendus de l'Académie des Sciences]]
 | pages = 616–618
 | title = Sur les polyèdres rationnels homothétiques à ''n'' dimensions
 | volume = 254
 | year = 1962
 | mr=0130860}}</ref>

===Reciprocity property===

The Ehrhart polynomial of the [[interior (topology)|interior]] of a closed convex polytope {{math|''P''}} can be computed as:

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

where {{math|''d''}} is the dimension of {{math|''P''}}. This result is known as Ehrhart–Macdonald reciprocity.<ref>Ehrhart, Eugène (1967), "Démonstration de la loi de réciprocité du polyèdre rationnel", ''Comptes Rendus de l'Academie des Sciences de Paris, Sér. A-B'' 265, A91–A94.</ref><ref>{{citation| last=Macdonald |first =Ian G.|authorlink=Ian G. Macdonald| title=Polynomials associated with finite cell-complexes|journal=[[Journal of the London Mathematical Society]]|year=1971|volume=2|issue=1|pages=181–192|doi=10.1112/jlms/s2-4.1.181 }}</ref>