ANNALES DE L'INSTITUT FOURIER

Équations fonctionnelles du dilogarithme  [ Functional equations for Rogers dilogarithm ]
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, p. 151-169
This paper proves a “new” family of functional equations $\left({\mathrm{Eq}}_{n}\right)$ for Rogers dilogarithm. These equations rely on the combinatorics of dihedral coordinates on moduli spaces of curves of genus $0$, ${ℳ}_{0,n}$. For $n=4$ we find back the duality relation while $n=5$ gives back the $5$ terms relation. It is then proved that the whole family reduces to the $5$ terms relation. In the author’s knownledge, it is the first time that an infinite family of functional equations for the dilogarithm with an increasing number of variables ($n-3$ for $\left({\mathrm{Eq}}_{n}\right)$) is reduced to the $5$ terms relation.This reduction explains the quotation marks around “new” at the beginning of this abstract.
Cet article démontre une « nouvelle » famille d’équations fonctionnelles $\left({\mathrm{Eq}}_{n}\right)$ ($n⩾4$) satisfaites par le dilogarithme de Rogers. Ces équations fonctionnelles reflètent la combinatoire des coordonnées diédrales des espaces de modules de courbes de genres $0$, ${ℳ}_{0,n}$. Pour $n=4$, on retrouve la relation de dualité et, pour $n=5$, la relation à $5$ termes du dilogarithme. Dans une seconde partie, on démontre que la famille $\left({\mathrm{Eq}}_{n}\right)$ se réduit à la relation à $5$ termes. C’est, à la connaissance de l’auteur, la première fois qu’une famille infinie d’équations fonctionnelles du dilogarithme ayant un nombre croissant de variables ($n-3$ pour $\left({\mathrm{Eq}}_{n}\right)$) se réduit à la relation à $5$ termes.La réduction de cette famille d’équations à la relation de $5$-cycle explique les guillemets de la première phrase.
Revised : 2017-05-28
Accepted : 2017-09-14
Published online : 2018-04-18
DOI : https://doi.org/10.5802/aif.3155
Classification:  39B50,  11G55,  39B22,  39B32,  33B30
Keywords: functional equations, dilogarithm, polylogarithms, moduli spaces, modulispaces of curves of genus 0
@article{AIF_2018__68_1_151_0,
author = {Soud\`eres, Ismael},
title = {\'Equations fonctionnelles du dilogarithme},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {68},
number = {1},
year = {2018},
pages = {151-169},
doi = {10.5802/aif.3155},
language = {fr},
url = {https://aif.centre-mersenne.org/item/AIF_2018__68_1_151_0}
}
Équations fonctionnelles du dilogarithme. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 151-169. doi : 10.5802/aif.3155. https://aif.centre-mersenne.org/item/AIF_2018__68_1_151_0/

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