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In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane. These polynomials are named after Eugène Ehrhart who studied them in the 1960s. ==Definition== Informally, if ''P'' is a polytope, and ''tP'' is the polytope formed by expanding ''P'' by a factor of ''t'' in each dimension, then ''L''(''P'', ''t'') is the number of integer lattice points in ''tP''. More formally, consider a lattice ''L'' in Euclidean space R''n'' and a ''d''-dimensional polytope ''P'' in R''n'' with the property that all vertices of the polytope are points of the lattice. (A common example is ''L'' = Z''n'' and a polytope for which all vertices have integer coordinates.) For any positive integer ''t'', let ''tP'' be the ''t''-fold dilation of ''P'' (the polytope formed by multiplying each vertex coordinate, in a basis for the lattice, by a factor of ''t''), and let : be the number of lattice points contained in the polytope ''tP''. Ehrhart showed in 1962 that ''L'' is a rational polynomial of degree ''d'' in ''t'', i.e. there exist rational numbers ''a''0,...,''a''''d'' such that: : for all positive integers ''t''. The Ehrhart polynomial of the interior of a closed convex polytope ''P'' can be computed as: : where ''d'' is the dimension of ''P''. This result is known as Ehrhart-Macdonald reciprocity. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ehrhart polynomial」の詳細全文を読む スポンサード リンク
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