Additive polylogarithms and their functional equations

Abstract

Let ${k[\varepsilon]_{2}:=k[\varepsilon]/(\varepsilon^{2})}$ . The single valued real analytic n-polylogarithm ${\mathcal{L}_{n}: \mathbb{C} \to \mathbb{R}}$ 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 ${li_{n}:k[\varepsilon]_{2}\to k}$ and prove that they satisfy functional equations analogous to those of ${\mathcal{L}_{n}}$. Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group ${B_{n}' (k[\varepsilon]_{2})}$ 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 k[ε]₂.