Independence of hyperlogarithms over function fields via algebraic combinatorics

TL;DR

This paper establishes a precise criterion for the linear independence of hyperlogarithm solutions of certain differential equations, emphasizing the role of Fuchsian-type equations and extending independence properties to broader coefficient rings.

Contribution

It provides a necessary and sufficient condition for hyperlogarithm independence and extends this property to the largest known ring of coefficients.

Findings

Linear independence characterized by Fuchsian-type equations

Extension of independence property to broader coefficient rings

Implication for algebraic combinatorics and differential equations

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 $dS=MS$) 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.