Effective Results and Methods for Diophantine Equations over Finitely Generated Domains

"This book is devoted to Diophantine equations where the solutions are taken from an integral domain of characteristic 0 that is finitely generated over Z, that is a domain of the shape Z[z1; :: zr] with quotient field of characteristic 0, where the generators z1; :: zr may be algebraic or transcendental over Q. For instance, the ring of integers and the rings of S-integers of a number field are finitely generated domains where all generators are algebraic. Our aim is to prove effective finiteness results for certain classes of Diophantine equations, i. e. , results that not only show that the equations from the said classes have only finitely many solutions, but whose proofs provide methods to determine the solutions in principle"-