2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59716We investigate a connection between the differential of polylogarithms (as considered by Cathelineau) and a finite variant of them. This allows to answer a question raised by Kontsevich concerning the construction of functional equations for the finite analogs, using in part the p-adic version of polylogarithms and recent work of Besser. Kontsevich's original unpublished note is supplied (with his kind permission) in an "Appendix" at the end of the paper.LaTeX, 44 pages, 1 figure (ps-version strongly recommended). Paper by Elbaz-Vincent and Gangl, an appendix by Kontsevich.K-Theory and HomologyNumber Theory19F15; 11T99; 33B99text