This work is a collaboration with B. Adamczewski (ICJ, France), T. Dreyfus (IRMA, France) and M. Wibmer (Graz University of Technology, Austria). In this talk, we will consider pairs of automorphisms $(\phi,\sigma)$ acting on fields of Laurent or Puiseux series: pairs of shift operators, of $q$-difference operators and of Mahler operators. Assuming that the operators $\phi$ and $\sigma$ are "independent", we show that their solutions are also "independent" in the sense that a solution $f$ to a linear $\phi$-equation and a solution $g$ to a linear $\sigma$-equation are algebraically independent over the field of rational functions unless one of them is a rational function. 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 questions and is based on a suitable Galois theory: the $\sigma$-Galois theory of linear $\phi$-equations developed by Ovchinnikov and Wibmer.