Multiple Polylogarithms: An Introduction
Multiple Polylogarithms: An Introduction
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多重多对数：简介
DOI：
10.1007/978-93-86279-10-1_1
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
Michel Waldschmidt
中科院分区：
文献类型：
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作者：
Michel Waldschmidt
当1，.，斯卡雷是正整数，a是单位圆盘上的复数时，单变量多重多项式被定义为。Fork= 1，这是经典的多重对数Lis（z）。这些多重多项式也可以用迭代Chen积分和satisfyshuffle关系来定义。多变量多重多项式定义为si ≥ 1，|紫|< 1（1 ≤i≤k）， L i（ s 1，s k，s k ) ( z1，zk ）= ∑ n 1 > n 2 > n> n k ≥1 z 1 n 1 克拉斯诺达尔 n k n 1 S 1 克伦克 S K 它们不仅满足shuffle关系，而且还满足stuffle关系。当对一个变量atz= 1的填充关系和多个变量atz 1 =zk= 1的填充关系进行特殊化时，可以得到定义为fork，s1，.，sk（s1 ≥ 2）的正整数的多重Zeta值之间的线性或二次依赖关系.
Multiple polylogarithms in a single variable are defined by, whens1, … ,skare positive integers andza complex number in the unit disk. Fork= 1, this is the classical polylogarithm Lis(z). These multiple polylogarithms can be defined also in terms of iterated Chen integrals and satisfyshuffle relations. Multiple polylogarithms in several variables are defined forsi≥ 1 and |zi| < 1(1 ≤i≤k) by L i ( s 1 ,⋯, s k ) ( z 1 ,⋯ z k )= ∑ n 1 > n 2 >⋯> n k ≥1 z 1 n 1 ⋯ z k n k n 1 s 1 ⋯ n k s k , and they satisfy not only shuffle relations, but alsostuffle relations. When one specializes the stuffle relations in one variable atz= 1 and the stuffle relations in several variables atz1= ⋯ =zk= 1, one gets linear or quadratic dependence relations between the Multiple Zeta Valueswhich are defined fork,s1, … ,skpositive integers withs1≥ 2.The Main Diophantine Conjecturestates that one obtains in this way all algebraic relations between these MZV.