On the Local Structure of Mahler Systems

This paper is a 1st step in the direction of a better understanding of the structure of the so-called Mahler systems: we classify these systems over the field H of Hahn series over Q and with value group Q. As an application of (a variant of) our main result, we give an alternative proof of the following fact: if, for almost all primes p, the reduction modulo p of a given Mahler equation with coefficients in Q(z) has a full set of algebraic solutions over Fₚ(z), then the given equation has a full set of solutions in Q(z) (this is analogous to Grothendieck’s conjecture for differential equations).