Editing Ehrhart polynomial (section)

==Ehrhart quasi-polynomials==
Let {{math|''P''}} be a rational polytope. In other words, suppose

:<math>P = \left\{ x\in\R^d : Ax \le b\right\},</math>

where <math>A \in \Q^{k \times d}</math> and <math>b \in \Q^k.</math> (Equivalently, {{math|''P''}} is the [[convex hull]] of finitely many points in <math>\Q^d.</math>) Then define

:<math>L(P, t) = \#\left(\left\{x\in\Z^d : Ax \le tb \right\} \right). </math>

In this case, {{math|''L''(''P'', ''t'')}} is a [[quasi-polynomial]] in {{math|''t''}}. Just as with integral polytopes, Ehrhart–Macdonald reciprocity holds, that is,

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

===Examples of Ehrhart quasi-polynomials===
Let {{math|''P''}} be a polygon with vertices (0,0), (0,2), (1,1) and ({{sfrac|3|2}}, 0). The number of integer points in {{math|''tP''}} will be counted by the quasi-polynomial <ref name=MR2271992>{{citation
 | last1 = Beck | first1 = Matthias
 | last2 = Robins | first2 = Sinai
 | mr = 2271992
 | isbn = 978-0-387-29139-0
 | location = New York
 | pages = [https://archive.org/details/computingcontinu00beck_082/page/n61 46]–47
 | publisher = Springer-Verlag
 | series = [[Undergraduate Texts in Mathematics]]
 | title = Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra
 | title-link = Computing the Continuous Discretely
 | year = 2007}}</ref>

: <math> L(P, t) = \frac{7t^2}{4} + \frac{5t}{2} + \frac{7 + (-1)^t}{8}. </math>