Unit Equations in Diophantine Number Theory

Unit equations are deceivingly simple polynomial equations of the form \[a_1x_1+\cdots+a_nx_n=1,\] where the coefficients \(a_i\) are non zero elements of a number field \(K\). One looks for solutions \(x_1,\ldots,x_n\) of these unit equations in the subgroup of units of a ring of \(S\)-integers of the field \(K\), where \(S\) is a finite set of places of \(K\), usually containing all infinite places.

Perhaps the first time one encounters a unit equation is for one proof of a theorem of Siegel (that a hyperelliptic curve over a number field \(K\) has only finitely many \(S\)-integral points) where the unit equation is of the form \(ax+by=1\), and one first proves that this unit equation has only finitely many solutions in \(S\)-units.

As such, unit equations are used as auxiliary equations to solve more general Diophantine problems, and one encounters these equations in most textbooks or monographs on Diophantine geometry or Diophantine analysis. Depending on the type of applications considered, one may be interested not only in determining if there are only finitely many solutions to a given unit equation, but also in finding effective bounds for the number of solutions. For example, Silverman’s The Arithmetic of Elliptic Curves (Springer, 2009, Second Edition), page 284, quotes the theorem, that for the unit equation \(ax+by=1\) one has the bound \(3\times 7^{[K: {\mathbb Q}]+2|S|}\) for the number of \(S\)-integer solutions of this unit equation, where \([K:{\mathbb Q}]\) is the degree of the number field \(K\) and \(|S|\) is the cardinality of the set of places \(S\) (J.-H. Evertse, “On equations in \(S\)-units and the Thue-Mahler equation”, Inv. Math 75 (3): 561–584, 1984).

Since Evertse is one the authors of the book under review, this is a good place to start. To begin with, the main objective of the book is to collect in one place the most important results on unit equations over number fields, especially finiteness results, effective bounds on the number of solutions, when available, and algorithms to find those solutions, for the case of unit equations in two unknowns. Additionally, the authors also treat the analog equations for function fields on one variable over an algebraically closed field of characteristic zero.

The book is almost self-contained: The first part, chapters 1, 2 and 3, gives a quick summary of the basic facts on algebraic number theory (traces, norms, discriminants, absolute values and places, rings of integers, \(S\)-integers, units and heights), algebraic function fields, and some results from Diophantine analysis. The second part is the core of the book, and in chapters 4, 5 and 6, treats the case of unit equations for number fields. The treatment is almost exhaustive, and one must be careful to avoid being lost in the many involved calculations of bounds. The occasional remark or aside may help to place several results in a more general context, such as the abc conjecture (a homogenized unit equation) and its several important consequences. The remaining chapters treat the analogues for function fields of transcendence degree one over an algebraically closed field and their finitely generated extensions, and some applications such as prime factors of sums of integers or dynamical systems defined by polynomials or rational maps.

Although addressed to the specialist, the book can be read by a graduate student working in this field, and fulfills its goal of gathering in one place most of the results on effective bounds for sets of solutions of unit equations.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx