On systems of equations over free products of groups

Using an analogue of the Makanin–Razborov diagrams, we give a description of the solution set of systems of equations over an equationally Noetherian free product of groups G. Equivalently, we give a parametrisation of the set Hom(H,G) of all homomorphisms from a finitely generated group H to G. Furthermore, we show that every algebraic set over G can be decomposed as a union of finitely many images of algebraic sets of NTQ systems.If the universal Horn theory of G (the theory of quasi-identities) is decidable, then our constructions are effective.