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30 pages, 4 figuresIn this paper, we study local solutions F=(F1,..,Fn) of a general functional equation of the form F1(U1(x,y))+....+Fn(Un(x,y))=0. A such equation will be called an ``abelian functional equation'' (Afe). We will restrict ourselves to the case when the inner functions Ui's are real rational functions. First we prove that if the components Fi of F are measurable then they are analytic and are characterized as solutions of a certain linear differential equation constructed from the Ui's. Then we apply our results to the resolution of Afe's which are generalised versions of the classical functionals equations satisfied by low order polylogarithms. In the last section, these results, interpreted in the frame of web geometry, give us new exceptional webs, related to configurations of points in CP2. Finally, we study the problem of characterizing the dilogarithm (resp. the trilogarithm) by Rogers (resp. Spence-kummer) functional equation

Codimension one webs are configurations of finitely many codimension one foliations in general position. Much of the classical theory evolved around the concept of abelian relation: a functional relation among the first integrals of the foliations defining the web reminiscent of Abel's addition theorem in classical algebraic geometry. The abelian relations of a given web form a finite dimensional vector space with dimension (the rank of the web) bounded by Castelnuovo number π(n, k) where n is the dimension of the ambient space and k is the number of foliations defining the web. A fundamental problem in web geometry is the classification of exceptional webs, that is, webs of maximal rank not equivalent to the dual of a projective curve. Recently, J.-M. Trépreau proved that there are no exceptional k-webs for n ≥ 3 and k ≥ 2n. In dimension two there are examples for arbitrary k and the classification problem is wide open.In this paper, we classify the exceptional Completely Decomposable Quasi-Linear (CDQL) webs globally defined on compact complex surfaces. By definition, the CDQL (k + 1)-webs are formed by the superposition of k linear foliations and one non-linear foliation. For instance, we show that up to projective transformations there are exactly four countable families and thirteen sporadic exceptional CDQL webs on P 2 .

Abstract. We investigate the space of abelian relations of planar webs admitting infinitesimal automorphisms. As an application we construct 4k − 14 new algebraic families of global exceptional k-webs on the projective plane, for each k ≥ 5.