Independence of Hyperlogarithms over Function Fields via Algebraic Combinatorics

Abstract

We obtain a necessary and sufficient condition for the linear independence of solutions of differential equations for hyperlogarithms. The key fact is that the multiplier (i.e. the factor M in the differential equation dS = M S) has only singularities of first order (Fuchsian-type equations) and this implies that they freely span a space which contains no primitive. We give direct applications where we extend the property of linear independence to the largest known ring of coefficients.

Cluster algebras and the subalgebra constructibility of the seven-particle remainder function

“…Additional cluster-algebraic identities, including an identity involving Li 3 evaluated on the cluster X -coordinates of D 4 , are discussed in [2,63]. Fortunately, all identities between polylogarithms are trivialized (up to algebraic identities between symbol letters) by the symbol map, for generic indices a i and argument z [64][65][66]. This map can be defined in terms of derivatives; for instance, taking the total derivative of (3.3), we have…”

About Some Drinfel’d Associators

“…In a previous work [3], we proved that asymptotic group-likeness, for a series, implies 5 that the series in question is group-like everywhere. The process above (theorem (1), Picard's process) can be performed, under certain conditions with improper integrals we then construct the series L recursively as…”

Mathematical Renormalization in Quantum Electrodynamics via Noncommutative Generating Series

“…• The polylogarithms form a basis of an infinite dimensional universal Picard-Vessiot extension by means of these differential equations [42,12] and their algebra, isomorphic to the shuffle algebra, admits {Li l } l∈L ynX as a transcendence basis, • The harmonic sums generate the coefficients of the ordinary Taylor expansions of their solutions (when these expansions exist) [43] and their algebra is isomorphic to the quasi-shuffle algebra admitting {H l } l∈L ynY as a transcendence basis, • The polyzetas do appear as the fondamental arithmetical constants involved in the computations of the monodromies [39,36], the Kummer type functional equations [40,36], the asymptotic expansions of solutions [43,44] and their algebra is freely generated by the polyzetas encoded by irreducible Lyndon words [44].…”

“…The definition of polylogarithms is extended over the words w ∈ X * by putting Li x 0 (z) := log(z). The {Li w } w∈X * are C -linearly independent [12,39,36] and then the following function, for v = y s 1 . .…”