  This paper is the first in a sequence on the first order theory of free
products. In the first paper we generalize the analysis of systems of equations
over free and (torsion-free) hyperbolic groups, and analyze system of equations
over free products. To do that we introduce limit groups over the class of free
products, and show that a finitely presented group has a canonical (finite)
collection of maximal limit quotients. We further extend this finite collection
and associate a Makanin-Razborov diagram over free products with a finitely
presented group. This MR diagram encodes all the quotients of a given finitely
presented group that are free products, all its homomorphisms into free
products, and equivalently all the solutions to a given system of equations
over a free product.
