翻訳と辞書 Words near each other ・ Lehmannia melitensis ・ Lehmannia nyctelia ・ Lehmann–Scheffé theorem ・ Lehmanotus ・ Lehmber Hussainpuri ・ Lehmbruck ・ Lehmbruck Museum ・ Lehmen ・ Lehmer ・ Lehmer code ・ Lehmer matrix ・ Lehmer mean ・ Lehmer number ・ Lehmer random number generator ・ Lehmer sieve ・ Lehmer's conjecture ・ Lehmer's GCD algorithm ・ Lehmer's totient problem ・ Lehmer–Schur algorithm ・ Lehmja ・ Lehmkuhl ・ Lehmkuhlen ・ Lehmo ・ Lehmon Colbert ・ Lehmon Pallo -77 ・ Lehmrade ・ Lehmstedt–Tanasescu reaction ・ Lehn House ・ Lehna Singh Majithia ・ Lehnar submachine gun Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lehmer's conjecture ： ウィキペディア英語版 Lehmer's conjecture Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts that there is an absolute constant $\backslash mu>1$ such that every polynomial with integer coefficients $P(x)\backslash in\backslash mathbb()$ satisfies one of the following properties: * The Mahler measure $\backslash mathcal(P(x))$ of $P(x)$ is greater than or equal to $\backslash mu$. * $P(x)$ is an integral multiple of a product of cyclotomic polynomials or the monomial $x$, in which case $\backslash mathcal(P(x))=1$. (Equivalently, every complex root of $P(x)$ is a root of unity or zero.) There are a number of definitions of the Mahler measure, one of which is to factor $P(x)$ over $\backslash mathbb$ as :$P(x)=a\_0\; (x-\backslash alpha\_1)(x-\backslash alpha\_2)\backslash cdots(x-\backslash alpha\_D),$ and then set :$\backslash mathcal(P(x))\; =\; |a\_0|\; \backslash prod\_^\; \backslash max(1,|\backslash alpha\_i|).$ The smallest known Mahler measure (greater than 1) is for "Lehmer's polynomial" :$P(x)=\; x^+x^9-x^7-x^6-x^5-x^4-x^3+x+1\; \backslash ,,$ for which the Mahler measure is the Salem number :$\backslash mathcal(P(x))=1.176280818\backslash dots\; \backslash \; .$ It is widely believed that this example represents the true minimal value: that is, $\backslash mu=1.176280818\backslash dots$ in Lehmer's conjecture.〔Smyth (2008) p.324〕 == Motivation == Consider Mahler measure for one variable and Jensen's formula shows that if $P(x)=a\_0\; (x-\backslash alpha\_1)(x-\backslash alpha\_2)\backslash cdots(x-\backslash alpha\_D)$ then :$\backslash mathcal(P(x))\; =\; |a\_0|\; \backslash prod\_^\; \backslash max(1,|\backslash alpha\_i|).$ In this paragraph denote　$m(P)=\backslash log(\backslash mathcal(P(x))$ , which is also called Mahler measure. If $P$ has integer coefficients, this shows that $\backslash mathcal(P)$ is an algebraic number so $m(P)$ is the logarithm of an algebraic integer. It also shows that $m(P)\backslash ge0$ and that if $m(P)=0$ then $P$ is a product of cyclotomic polynomials i.e. monic polynomials whose all roots are roots of unity, or a monomial polynomial of $x$ i.e. a power $x^n$ for some $n$ . Lehmer noticed〔David Boyd (1981). "Speculations concerning the range of Mahler's measure" Canad. Math. Bull. Vol. 24(4)〕 that $m(P)=0$ is an important value in the study of the integer sequences $\backslash Delta\_n=\backslash text(P(x),\; x^n-1)=\backslash prod^D\_(\backslash alpha\_i^n-1)$ for monic $P$ . If $P$ does not vanish on the circle then $\backslash lim|\backslash Delta\_n|^=\backslash mathcal(P)$ and this statement might be true even if $P$ does vanish on the circle. By this he was led to ask :whether there is a constant $c>0$ such that $m(P)>c$ provided $P$ is not cyclotomic?, or :given $c>0$, are there $P$ with integer coefficients for which Some positive answers have been provided as follows, but Lehmer's conjecture is not yet completely proved and is still a question of much interest. 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「Lehmer's conjecture」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.