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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Normal polytope ： ウィキペディア英語版 Normal polytope In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope ''P'' is called normal if it has the following property: given any positive integer ''n'', every lattice point of the dilation ''nP'', obtained from ''P'' by scaling its vertices by the factor ''n'' and taking the convex hull of the resulting points, can be written as the sum of exactly ''n'' lattice points in ''P''. This property plays an important role in the theory of toric varieties, where it corresponds to projective normality of the toric variety determined by ''P''. Normal polytopes have popularity in algebraic combinatorics. These polytopes also represent the homogeneous case of the Hilbert bases of finite positive rational cones and the connection to algebraic geometry is that they define projectively normal embeddings of toric varieties. ==Definition== Let ''P'' ⊂ ℝ''d'' be a lattice polytope. Denote the affine lattice in ℤ''d'' generated by the lattice points in ''P'' by ''L'': :$L=v+\backslash sum\_(x-y)\backslash subseteq\; \backslash mathbb^d$ where ''v'' is some lattice point in ''P''. P is integrally closed if the following condition is satisfied: :$c\backslash in\backslash mathbb,\; z\backslash in\; cP\backslash cap\backslash mathbb^d\backslash implies\; \backslash exists\; x\_1,\backslash ldots,x\_c\backslash in\; P\backslash cap\backslash mathbb^d$ such that $x\_1+\backslash cdots+x\_c=z$. ''P'' is normal if the following condition is satisfied: :$c\backslash in\; \backslash mathbb,\; z\backslash in\; cP\backslash cap\; L\backslash implies\; \backslash exists\; x\_1,\backslash ldots,x\_c\backslash in\; P\backslash cap\; L$ such that $x\_1+\backslash cdots+x\_c=z$. The normality property is invariant under affine-lattice isomorphisms of lattice polytopes and the integrally closed property is invariant under an affine change of coordinates. Note sometimes in combinatorial literature the difference between normal and integrally closed is blurred. 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「Normal polytope」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.