Algebraic independence and linear difference equations

We consider pairs of automorphisms (0, a) acting on fields of Laurent or Puiseux series: a: x 7! q2x), and of Mahler operators (0: x 7! xp1 , a: x 7! xp2). Given a solution f to a linear 0-equation and a solution g to an algebraic a-equation, both transcendental, we show that f and g are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of q-hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the a-Galois theory of linear 0-equations.