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Additive polylogarithms and their functional equations
Authors:Sinan Ünver
Institution:1. Department of Mathematics, Ko? University, Istanbul, Turkey
Abstract:Let ${k\varepsilon]_{2}:=k\varepsilon]/(\varepsilon^{2})}Let ke]2:=ke]/(e2){k\varepsilon]_{2}:=k\varepsilon]/(\varepsilon^{2})} . The single valued real analytic n-polylogarithm Ln: \mathbbC ? \mathbbR{\mathcal{L}_{n}: \mathbb{C} \to \mathbb{R}} is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in ünver (Algebra Number Theory 3:1–34, 2009) to define additive n-polylogarithms lin:ke]2? k{li_{n}:k\varepsilon]_{2}\to k} and prove that they satisfy functional equations analogous to those of Ln{\mathcal{L}_{n}}. Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group Bn¢(ke]2){B_{n}' (k\varepsilon]_{2})} defined by Goncharov (Adv Math 114:197–318, 1995). We hope that these functions will be useful in the study of weight n motivic cohomology over kε]2.
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