Abstract

Here we characterise, in a complete and explicit way, the relations of algebraic dependence over ${\mathbb{Q}}$ of complex values of Hecke-Mahler series taken at algebraic points $\underline{u}_1,\ldots,\underline{u}_m$ of the multiplicative group ${\mathbb{G}}_{{\rm m}}^2$, under a technical hypothesis that a certain sub-module of ${\mathbb{G}}_{{\rm m}}^2$ generated by the $\underline{u}_i$’s has rank one. This is the first part of a work, announced in [Pel1], whose main objective is completely to solve a general problem on the algebraic independence of values of these series