Équations fonctionnelles du dilogarithme
[Functional equations for Rogers dilogarithm]
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 151-169.

This paper proves a “new” family of functional equations (Eq n ) for Rogers dilogarithm. These equations rely on the combinatorics of dihedral coordinates on moduli spaces of curves of genus 0, 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 (Eq n )) 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 (Eq n ) (n4) satisfaites par le dilogarithme de Rogers. Ces équations fonctionnelles reflètent la combinatoire des coordonnées diédrales des espaces de modules de courbes de genres 0, 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 (Eq n ) 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 (Eq n )) 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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3155
Classification: 39B50,  11G55,  39B22,  39B32,  33B30
Keywords: functional equations, dilogarithm, polylogarithms, moduli spaces, modulispaces of curves of genus 0
Soudères, Ismael 1

1 Universität Osnabrück Institut für Mathematik Albrechtstr. 28a DE-49076 Osnabrück (Germany)
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Soudères, Ismael. 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/articles/10.5802/aif.3155/

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