Diophantine equations are systems of polynomial equations to be solved in
integers, in rational numbers, or in various generalizations, such as finitely
generated rings over Z or finitely generated fields over Q. Diophantine
approximation is the study of Diophantine equations using the method of
approximations. The Nevanlinna theory, on the other hand, studies holo-
morphic solutions of the systems of polynomial equations. More precisely,
since the complex solutions to a system of polynomial equations form an
algebraic variety, Diophantine approximation studies the rational points in
algebraic varieties defined over Q and Nevanlinna theory investigates the
properties of holomorphic curves in algebraic varieties over C. Nevanlinna
theory and Diophantine approximation have developed independently of
one another for several decades. It has been, however, discovered by C.F.
Osgood, P. Vojta, Serge Lang and others that a number of striking similar-
similarities exist between these two subjects. Generally speaking, a non-constant
holomorphic curve in an algebraic variety corresponds to an infinite set of
rational points, so, in this way, any theorem in Nevanlinna theory should
translate into a true statement in Diophantine approximation. A growing
understanding of these connections over the last 15 years has led to signif-
significant advances in both fields. Outstanding conjectures from decades ago
are being solved.