Diophantine number theory (the study of Diophantine equations, Diophantine inequalities and their applications) is a very active area in number theory with a long history. This book is about unit equations, a class of Diophantine equations of central importance in Diophantine number theory, and their applications. Unit equations are equations of the form

a1x1 +… + anxn = 1

to be solved in elements x1, …, xn from a finitely generated multiplicative group Γ, contained in a field K, where a1, …, an are non-zero elements of K. Such equations were studied originally in the cases where the number of unknowns n = 2, K is a number field and Γ is the group of units of the ring of integers of K, or more generally, where Γ is the group of S-units in K. Unit equations have a great variety of applications, among others to other classes of Diophantine equations, to algebraic number theory and to Diophantine geometry.

Certain results concerning unit equations and their applications covered in our book were already presented, mostly in special or weaker form, in the books of Lang (1962, 1978, 1983), Győry (1980b), Sprindžuk (1982, 1993), Evertse (1983),Mason (1984), Shorey and Tijdeman (1986), deWeger (1989), Schmidt (1991), Smart (1998), Bombieri and Gubler (2006), Baker and Wüstholz (2007) and Zannier (2009), and in the survey papers of Evertse, Győry, Stewart and Tijdeman (1988b), Győry (1992a, 1996, 2002a, 2010) and Bérczes, Evertse and Győry (2007b).

In 1988, we wrote, together with Stewart and Tijdeman, the survey Evertse, Győry, Stewart and Tijdeman (1988b) on unit equations and their applications giving the state of the art of the subject at that time. Since then, the theory of unit equations has been greatly expanded. In the present book we have tried to give a comprehensive and up-to-date treatment of unit equations and their applications. We prove effective finiteness results for unit equations in two unknowns, describe practical algorithms to solve such equations, give explicit upper bounds for the number of solutions, discuss analogues of unit equations over function fields and over finitely generated domains, and present various applications.