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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 Restricted sumset ： ウィキペディア英語版 Restricted sumset In additive number theory and combinatorics, a restricted sumset has the form :$S=\backslash ,$ where $A\_1,\backslash ldots,A\_n$ are finite nonempty subsets of a field ''F'' and $P(x\_1,\backslash ldots,x\_n)$ is a polynomial over ''F''. When $P(x\_1,\backslash ldots,x\_n)=1$, ''S'' is the usual sumset $A\_1+\backslash cdots+A\_n$ which is denoted by ''nA'' if $A\_1=\backslash cdots=A\_n=A$; when :$P(x\_1,\backslash ldots,x\_n)=\backslash prod\_(x\_j-x\_i),$ ''S'' is written as $A\_1\backslash dotplus\backslash cdots\backslash dotplus\; A\_n$ which is denoted by $n^\; A$ if $A\_1=\backslash cdots=A\_n=A$. Note that |''S''| > 0 if and only if there exist $a\_1\backslash in\; A\_1,\backslash ldots,a\_n\backslash in\; A\_n$ with $P(a\_1,\backslash ldots,a\_n)\backslash not=0$. == Cauchy-Davenport theorem == The Cauchy–Davenport theorem named after Augustin Louis Cauchy and Harold Davenport asserts that for any prime ''p'' and nonempty subsets ''A'' and ''B'' of the prime order cyclic group Z/''p''Z we have the inequality〔Nathanson (1996) p.44〕〔Geroldinger & Ruzsa (2009) pp.141–142〕 :$|A+B|\backslash ge\backslash min\backslash .\backslash ,$ We may use this to deduce the Erdős–Ginzburg–Ziv theorem: given any sequence of 2''n''−1 elements in Z/''n'', there are ''n'' elements that sums to zero modulo ''n''. (Here ''n'' does not need to be prime.)〔Nathanson (1996) p.48〕〔Geroldinger & Ruzsa (2009) p.53〕 A direct consequence of the Cauchy-Davenport theorem is: Given any set ''S'' of ''p''−1 or more elements, not necessarily distinct, of Z/''p''Z, every element of Z/''p''Z can be written as the sum of the elements of some subset (possibly empty) of ''S''.〔Wolfram's MathWorld, Cauchy-Davenport Theorem, http://mathworld.wolfram.com/Cauchy-DavenportTheorem.html, accessed 20 June 2012.〕 Kneser's theorem generalises this to finite abelian groups.〔Geroldinger & Ruzsa (2009) p.143〕 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「Restricted sumset」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.