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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 Bernstein–Kushnirenko theorem ： ウィキペディア英語版 Bernstein–Kushnirenko theorem Bernstein–Kushnirenko theorem (also known as BKK theorem or Bernstein–Khovanskii–Kushnirenko theorem 〔 *David A. Cox; J. Little; D. O'Shea Using algebraic geometry. Second edition. Graduate Texts in Mathematics, 185. Springer, 2005. xii+572 pp. ISBN 0-387-20706-6〕), proven by David Bernstein 〔D. N. Bernstein, "The number of roots of a system of equations", ''Funct. Anal. Appl.'' 9 (1975), 183–185〕 and Anatoli Kushnirenko 〔A. G. Kouchnirenko, "Polyhedres de Newton et nombres de Milnor", ''Invent. Math.'' 32 (1976), 1–31〕 in 1975, is a theorem in algebra. It claims that the number of non-zero complex solutions of a system of Laurent polynomial equations ''f''1 = 0, ..., ''f''''n'' = 0 is equal to the mixed volume of the Newton polytopes of ''f''1, ..., ''f''''n'', assuming that all non-zero coefficients of ''fn'' are generic. More precise statement is as follows: ==Theorem statement== Let $A$ be a finite subset of $\backslash mathbb^n$. Consider the subspace $L\_A$ of the Laurent polynomial algebra $L\_A\; =\; \backslash ,$ where $c\_\backslash alpha\; \backslash in\; \backslash mathbb$ and for each $\backslash alpha\; =\; (a\_1,\; \backslash ldots,\; a\_n)\; \backslash in\; \backslash mathbb^n$ we have used the shorthand notation $x^\backslash alpha$ to write the monomial $x\_1^\; \backslash cdots\; x\_n^$. Now take $n$ finite subsets $A\_1,\; \backslash ldots,\; A\_n$ with the corresponding subspaces of Laurent polynomials $L\_,\; \backslash ldots,\; L\_$. Consider a generic system of equations from these subspaces, that is: $f\_1(x)\; =\; \backslash ldots\; =\; f\_n(x)\; =\; 0,$ where each $f\_i$ is a generic element in the (finite dimensional vector space) $L\_$. The Bernstein–Kushnirenko theorem states that the number of solutions $x\; \backslash in\; (\backslash mathbb\; \backslash setminus\; 0)^n$ of such a system is equal to $n!\; V(\backslash Delta\_1,\; \backslash ldots,\; V\_n)$, where $V$ denotes the Minkowski mixed volume and for each $i$, $\backslash Delta\_i$ is the convex hull of the finite set of points $A\_i$. Clearly $A\_i$ is a convex lattice polytope. It can be interpreted as the Newton polytope of a generic element of generic element of the subspace $L\_$. In particular, if all the sets $A\_i$ are the same $A\; =\; A\_1\; =\; \backslash cdots\; =\; A\_n$, then the number of solutions of a generic system of Laurent polynomials from $L\_A$ is equal to $n!\; vol(\backslash Delta)$ where $\backslash Delta$ is the convex hull of $A$ and vol is the usual $n$-dimensional Euclidean volume. Note that even though the volume of a lattice polytope is not necessarily an integer but it is an integer after multiplying by $n!$. 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「Bernstein–Kushnirenko theorem」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.