We 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 Kontsevich's original unpublished note is supplied (with his kind permission) in an ‘Appendix’ at the end of the paper.
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Elbaz–Vincent et al. (2002) studied this question.
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