Existential questions in (relatively) hyperbolic groups

We study the decidability of the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. Our tools are Rips and Sela’s canonical representatives for these groups, and solvability of equations with rational constraints (involving finite state automata) in free groups and free products.