Abelian functional equations, planar web geometry and polylogarithms

Abstract.

We study some abelian functional equations (Afe). They are equations in the F _i ’s of the form F ₁(U ₁) + ... + F _N (U _N ) = 0 where the U _i ’s are real rational functions in two variables. First we prove that the local measurable solutions are actually analytic and we characterize their components as solutions of linear differential equations constructed from the U _i ’s. Then we propose two methods for solving Afe. Next we apply these methods to the explicit resolution of generalized versions of classical (inhomogeneous) Afe satisfied by low order polylogarithms. Interpreted in the framework of web geometry, these results give us new nonlinearizable maximal rank planar webs. Then we observe that there is a relation between these webs and certain configurations of points in $\mathbb{C}\mathbb{P}^{2} ,$ which leads us to define the notion of web associated to a configuration: these webs seem of high rank and could provide numerous new exceptional webs. Finally, we use the preceding results to show that, under weak regularity assumptions, the trilogarithm is the only function which satisfies the Spence–Kummer equation.