On poly(ana)logs. I. (With an appendix by Maxim Kontsevich: The \(1\frac{1}{2}\)-logarithm)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1062.11042
DOI: 10.1023/A:1013757217319
author: Philippe Elbaz-Vincent
author: Herbert Gangl
published in: Compositio Mathematica
publication date: 2002-04-11T00:00:00Z
Mathematics Subject Classification ID: 11G55
Mathematics Subject Classification ID: 19F15
Mathematics Subject Classification ID: 33B30
zbMATH DE Number: 1727234
zbMATH Keywords: polylogarithms
zbMATH Keywords: functional equations
zbMATH Keywords: derivations
zbMATH Keywords: Goncharov complexes
zbMATH Keywords: finite polylogarithms
zbMATH Keywords: infinitesimal polylogarithms
MaRDI profile type: Publication
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title: On poly(ana)logs. I. (With an appendix by Maxim Kontsevich: The \(1\frac{1}{2}\)-logarithm)
review text: The authors investigate functional properties of the infinitesimal polylogarithms and the relationship between finite, infinitesimal and usual polylogarithms. The paper ends with a reproduction of a note of Kontsevich on the 1\(\frac{1}{2}\)-logarithm.NEWLINENEWLINEThe \(n\)-th finite polylogarithm over a field \(F\) of characteristic \(p>0\) is the polynomial function \(\mathcal L_n(x) = \text{li}_n(x) = \sum_{k=1}^{p-1} x^k/k^n \in F[x]\). It was introduced for \(n=1\) by M. Kontsevich in an unpublished note (which is given as appendix of the present paper), in which he observed that the 1\(\frac{1}{2}\)-logarithm \(\mathcal L_1\) satisfies the same 4-term functional equation as the infinitesimal dilogarithm \(-(x\log|x| +(1-x)\log| 1-x|\). His explicit question for the next case was answered when it was found that the functional equation of J.-L. Cathelineau for the infinitesimal trilogarithm was also satisfied by \(\mathcal L_2\).NEWLINENEWLINEIn the paper under review, the authors investigate the relationship between finite, infinitesimal and usual polylogarithms and prove directely several polynomial identities, such as the 22-term functional equation satisfied by \(\mathcal L_2(x)\), which is obtained from the Cathelineau identity for the infinitesimal trilogarithm. As a matter of fact, the general machinery used to deduce the functional equation for higher \(\mathcal L_{n-1}\) from functional equations for infinitesimal \(n\)-logarithms relies on the following three main features:NEWLINENEWLINE(i) The fact that the \(p\)-adic polylogarithms introduced by \textit{R. Coleman} [Invent. Math. 69, 171--208 (1982; Zbl 0516.12017)] satisfy the same functional equations as the classical ones, as proved by \textit{Z. Wojtkowiak} [Bull. Soc. Math. Fr. 119, 343--370 (1991; Zbl 0748.12006)]. NEWLINENEWLINE(ii) The tangential procedure of \textit{J.-L. Cathelineau} [Ann. Inst. Fourier 46, 1327--1347 (1996; Zbl 0861.19003)], which yields an analogue for infinitesimal poly…
reviewed by: Jean-François Jaulent
description: scientific article; zbMATH DE number 1727234