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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 Mahler measure ： ウィキペディア英語版 Mahler measure In mathematics, the Mahler measure $M(p)$ of a polynomial $p(z)$ with complex coefficients is :$M(p)\; =\; |a|\backslash prod\_\; |\backslash alpha\_i|\; =\; |a|\; \backslash prod\_^n\; \backslash max\backslash ,$ where $p(z)$ factorizes over the complex numbers $\backslash mathbb$ as :$p(z)\; =\; a(z-\backslash alpha\_1)(z-\backslash alpha\_2)\backslash cdots(z-\backslash alpha\_n).$ It can be shown using Jensen's formula that also :$M(p)\; =\; \backslash exp\backslash left(\; \backslash frac\; \backslash int\_^\; \backslash ln(|p(e^)|)\backslash ,\; d\backslash theta\; \backslash right),$ which is the geometric mean of $|p(z)|$ for $z$ on the unit circle $|z|\; =\; 1.$ The Mahler measure of an algebraic number $\backslash alpha$ is defined as the Mahler measure of the minimal polynomial of $\backslash alpha$ over $\backslash mathbb$. In particular, if $\backslash alpha$ is a Pisot number or a Salem number then its Mahler measure is simply $\backslash alpha$. The Mahler measure is named after Kurt Mahler. It is in fact a kind of Height function. ==Properties== * The Mahler measure is multiplicative, i.e. $M(p\backslash ,q)\; =\; M(p)\; \backslash cdot\; M(q).$ * Also $M(p)\; =\; \backslash lim\_\; \backslash |p\backslash |\_,$ where : $\backslash |p\backslash |\_\backslash tau\; =\backslash left(\; \backslash frac\; \backslash int\_0^\; |p(e^)|^\backslash tau\; \backslash ,\; d\backslash theta\; \backslash right)^\; \backslash ,$ is the $L\_\backslash tau$ norm of $p$ (although this is not a true norm for values of ). * (Kronecker's Theorem) If $p$ is an irreducible monic integer polynomial with $M(p)\; =\; 1$, then either $p(z)\; =\; z,$ or $p$ is a cyclotomic polynomial. * Lehmer's conjecture asserts that there is a constant $\backslash mu>1$ such that if $p$ is an irreducible integer polynomial, then either $M(p)=1$ or $M(p)>\backslash mu$. * The Mahler measure of a monic integer polynomial is a Perron number. 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「Mahler measure」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.