Independence of hyperlogarithms over function fields via algebraic combinatorics

2011/01/24 by Matthieu Deneufchâtel, Deneufchâtel, Matthieu, Gérard Duchamp +5 · 4 citations
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Polynomial and algebraic computation #Symbolic Computation (cs.SC)

paper · doi:10.48550/arxiv.1101.4497

openalex publication_date 2011/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Three-loop banana integrals with three equal masses
• Hyperlogarithms: Functions on Free Monoids
• Three-loop banana integrals with four unequal masses
• The integro-differential closure of a commutative differential ring

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